I started on this as it looked like a pleasant naval mystery. I'm providing a Summary Answer above, and below my method for reaching that conclusion as well as other relevant information. A lot of my final version for this answer derives from a serendipitous find of a KuK Kriegsmarine document without which it would have been impossible to include this detail.

Answer

The Nassadista were crew-members of military riverine rowing boats under direct royal supervision and command. Their river vessels were primarily biremes with some one-rower vessels. The vessels had shallow aft- and forecastles, one or two masts for sails, and, after King Sigismund's time, cannon.

Reusner's—the most numerous primary source—mentions for the Nassadista are in the context of the 1529 Siege of Vienna. The troops were part of the Austrian/Hungarian forces (possibly on Italian-built ships in that case). However, the Nassadists had been in existence as a 'unit' who provided service to the King since the 14th century with special rights and privileges. Their river warfare skills were instrumental in the 15th century during the sieges of Nândor-Fehérvâr in 1440 and 1456.

Primary Sources

My synthesis of the primary sources is rather poor as I was limited by my understanding of Latin. Online translations, such as Google, of these are garbled nonsense.

Reusner

Nicolaus Reusner's 'Rerum memorabilium in Pannonia...' has three mentions, all in the context of the 1529 Siege of Vienna:

^{[p59] Porte vrbis omnes clause, communitaque fuere, vna Salis porta, quaad Danubium itur, aperta, vnde eruptio, si qui casus tulisset sieri posset. Nauigia bellica, celoces que optimi regis Ferdinandi prouidencia maximis impensis, à fabris Italis, aduersus Turcarum Nassadistas instructa, armata que in naualibus exusta, concisa ac submersa sunt. Naute enim Itali , qui in spe erant, frustra expectabantur, ad murum proxima edificia, quae gerendis rebus obstare videbantur, solo equata.}

The gist seems to be that Italian-made ships of war (ordered built by King Ferdinand) with the Nassadists issued against the Turks, set them on fire, and sunk them while a sally was organised by land forces.

^{[p64] De Turcae consilio interrogatus respondit: decreuisse vrbem, disiectis puluerum submotorum vi turribus, violenter derepente capere: docuit loca, quibus perfodere pataret : de tormentis, respondit : terra, quadringenta aenea tormenta aduecta esse, quae pugni magnitudine globos iacularentur, ac maiora etiam quaedam. Aduerso autem Danubio non plura subornata decem praegrandibus xx. pedum longitudine. De nauigiis Turcae respondit: Imbrahim Bassae sexaginta naues parere; Turcarum Caesaris nauium se numerum nescire. Nassadistarum autem quadringentos esse, omnibus porro nauigiis quinque armatorum rum millia asscripta: praegrandia quaedam, aliquor retro à Vienna milliaribus, relicta.}

The context of this part seems to be that a Turkish response involved Ibrahim Pasha in the lead of sixty ships; however, there were four hundred Nassadista, all supported by five soldiers [and a thousand conscripts?].

^{[p70] Postridie porro, Caesarem, qui cum exercitu praecesserat, persecutus ea celeritate, vt quinque diebus Budam venerint, quae XXXII praelongis Germanicis miliaribus à Vienna abest, innumeri equi ex fuga mortui corruerunt, nec non è Turcis, captiuisque Christianis passim in utis plurimi, cum assequi non postent, mortui reperti sunt. Paulus BAEFITSCH / Sigismundus MEIRELBERGER & Ioannes CAZIANER / leui armatura fugientes persecuti, multos ceperunt, plurimos interfecerunt, é captiuis etiam Christianis non paucos liberauerunt : Nassadistae quoq; ad Posonium ex vrbe arceq;, cui praefectus fuit Wolfgangus ODER / impugnaci quibusdam demersis nauibus magnam iacturam fecerut. Captiui Turcae pro certo affirmarut, Turcarum Cesarem maxima damna accepisse, tum quod plurimi ex illius militib.caesi igneariorumq; vi confecti sunt, multi fame enecti: ...}

In a chase of the Turks, some notables died but the troops ran down some of the Turks, and liberate Christian prisoners, of whom some were Nassadista.

Henricpetri

Henricpetri's 'Historicum opus, in quatuor tomos diuisum' copies Reusner word-by-word (p468) (Reusner's third citation above).

Ogier Ghiselin de Busbecq

Ogier Ghiselin de Busbecq's 'The Life and Letters of Ogier Ghiselin de Busbecq (Complete)' includes this mention:

On the next day we came to Gran, and proceeded [...] to Komorn, which is the river fortress of his Imperial Majesty, and stands on the river Waag. On either bank of the river the garrison of the place with the naval auxiliaries, who are there called Nassadistas, was awaiting us.

A straightforward statement claiming the Nassadistas were naval auxiliaries.

Secondary Sources

Glonung

^{NASSADIST 'Besatzungsmitglied einer Nassada, eines Schiffstyps'. Zu Nassada, Nassade. ...}

^{Militärische Fremdwörter spielen in diesem Text eine untergoerdnete Rolle: Janitschar und Nassadist als Personenbezeichnungen, ... Die Herkunft dieser Ausdrücke deutet zum einen aug den lange anderen auf die regionalen zusammenhänge bei den Gefechten mit den Türken in den Ostgebieten (Janitschar, Nassadist, Tschicka).}

_{^{—Glonung, 'Organisation und Entwicklung historischer Wortschätze'}}

Translated (Google Translate, modified by myself):

'Crew member of a Nassada, a type of ship'. To Nassada, Nassade. ...

Foreign military words play a subordinate role in this text: Janissary and Nassadist as personal names, ... The origin of these expressions on the one hand indicates the regional connections in the battles with the Turks in the eastern regions (Janissary, Nassadist, Tschicka).

The very useful input from this part is the region-specific connection to the Turkish wars.

Lajos

^{A csupán csak katonának (Kriegsmann) nevezettek mellett a német (Landsknecht) és magyar (Trabant) gyalogosok, a huszárok, naszádosok (Nassadist), a pattantyúsok (Büchsenmeister) tarka forgatagában számos altiszt, ...}

_{^{—Lajos, 'Gyir Eridváros Kiépítése A 16. Század Második Felében'}}

Translated (Google Translate, non-modified):

In addition to those who are only called soldiers (Kriegsmann), there are several non-commissioned officers in the variegated hustle and bustle of German (Landsknecht) and Hungarian (Trabant) infantry, hussars, Nassadists, and Büchsenmeister, ...

While adding some context, such as non-commissioned officers, Gecsényi Lajos also helpfully links the Hungarian "naszádosok" with the Latin "Nassadist". The Hungarian "naszádosok" is related to "naszád", a relatively wide term for a sailing ship in Hungarian, and translated as "Hungarian fluvial warfare boat" by some modelling enthusiasts. The "-os" and "-ok" suffixes denote "having something"/noun-forming (such as "table" to "carpenter") (from the linked WP) and plurality.

The Hungarian naszád represents a one- to two-sail riverine galley of between 12 to 24 oars with a shallow aft- and forecastle, and possibly equipped with a forward cannon as per the image:

Therefore, a translation of "naszádosok/Nassadist" as the crew linked to a naszád (riverine galley) seems appropriate. This matches how the term was used in the primary source quotations.

Thanks to @b.Lorenz for comments regarding this section!

Historiography

I serendipitously found a KuK Kriegsmarine document from 1888, 'Zur Geschichte der Donauflottille von den Römerzeiten bis zur Schlacht bei Mohacs 1526' which included additional useful detail. The below is based on that (not quoted further), all translations from German are via Google Translate modified by myself.

Origins

^{Ganz dieselben Privilegien und auch Verpflichtungen wie die Burgsassen, hatten die königlichen Schiffer, oder wie sie später genannt wurden, Nassadisten, welche in den an den Stromufern gelegenen Ortschaften und Kronländereien sesshaft waren. Es erhellt dies auch aus einem Documente des Jahres 1336, in welchem König Karl Robert unter anderem erklärt, »dass alle Privilegien der Schiffer aufrecht verbleiben, ebenso wie ihre V e r p f l i c h t u n g e n im Dienste des Königs, wie dies schon seit a l t e r s h er besteht. Die Nassadisten hatten gerade so wie die Burgsassen Kriegsdienste zu leisten und auch ihre Schiffe beizustellen; also eine ähnliche Institution, wie sie König Wilhelm der Eroberer in England in den cinq ports eingeführt hatte.—}

The royal boatmen, or as they were later called, Nassadists, had the same privileges and obligations as the castle residents, who were settled in the towns and crown lands on the banks of the river. This is also evident from a document from 1336, in which King Karl Robert explains, among other things, »that all the privileges of the sailors remain intact, as do their obligations in the service of the King, as has existed for a long time. The Nassadists, like the Burgsasses, had to do military service and also provide their ships; in other words, an institution similar to the one that King William the Conqueror had introduced to cinq ports in England.-

The document also describes how the Nassadists were instrumental in Johann Hunyadi's victories against Mahomet II at Nândor-Fehérvâr in 1440 and also during 1456. In general, the document describes the Nassadist's as more manoeuvrable than the Turkish fleet. After Mathias Corvinius, the fleet went into decline, but was still loyal and true as this description of the mid-1520's notes:

^{Jaicza in Bosnien stand noch, unterstützt von der dort stationierenden Nassadenflottille. Man muss die Treue der Nassadisten bewundern, wenn man liest, wie dieselben zu einer Zeit, als in Ungarn bereits alles zu wanken begann, hungernd und zerfetzt ihrem Dienste nachkamen und sich nicht verliefen, obwohl sie keinen Sold erhielten.}

Jaicza in Bosnia was still standing, supported by the Nassadenflotilla stationed there. One has to admire the loyalty of the Nassadists when one reads how at a time when everything was beginning to waver in Hungary they were starving and tattered and did not run, even though they were not paid.

Command Structure

^{An der Spitze der Flotte stand der »fövezer«, d. h. Oberstcommandierende. Später, nämlich im XV. Jahrhundert, hieß der Admiral Obercapitän. Derselbe war aber nicht immer der wirkliche Befehlshaber der Nassadeu, sondern diesen Rang erhielt, um das Corps auszuzeichnen, der Palatin oder ein anderer der höchsten Bannerherren, die auch den obersten Befehl über das Landheer führten. In diesem Falle war der Vice-Obercapitän der wirkliche Commandant. Den Befehl über größere oder kleinere A b t e i lungen der Flotte führten die Capitane. Die Commandanten der einzelnen Nassaden oder auch mehrerer kleineren Schiffe hießen Woiwoden. Diesen unterstanden drei oder auch mehrere Unterofficiere. Die mittelalterlichen lateinischen Documente nennen sie Decuriones. Dann gab es Schiff- und Waffenmeister, praefecti armamentarii, wie sie in den königlichen Decreteu genannt werden. Auf den kleineren Schiffen hatte der Obersteuermann auch die Aufsicht über die Waffen.}

At the head of the fleet was the "fövezer", ie. Colonel Commandant. Later, namely in the XV. century, he was called the Admiral Chief Captain. However, he was not always the real commander of the Nassadeu, but was given this rank in order to honor the corps, the Palatine or another of the highest lords, who also carried out the highest command over the land army. In this case the Vice-Captain was the real commandant. The captains commanded larger or smaller divisions of the fleet. The commandants of the individual Nassaden or several smaller ships were called voivodes. These were subordinate to three or more non-commissioned officers. The medieval Latin documents call them Decuriones. Then there were ship and weapon masters, praefecti armamentarii, as they are called in the royal decrees. The helmsman also oversaw the weapons on the smaller ships.

Armament

^{Die Bewaffnung der Nassadisten bestand aus Säbel, Spieß und Schild. ... In den Zeiten der Hunyaden bestand ein Theil der Besatzungen aus Lanzenträgern und schweren Fußtruppen, die Brustpanzer, Lederkoller oder Panzerhemd trugen. Die Schützen gebrauchten Pfeilbogen, später Armbrust. Die Musketiere bedienten die pissides manuales, schwere Musketen auf Ständern, welche wohl nicht viel Schaden anrichteten, auch nicht sehr beliebt waren.}

The arms of the Nassadists consisted of sabers, skewers and shields. ... In the Hunyad times, some of the crews consisted of lancers and heavy foot troops, who wore breastplates, leather bollards, or armoured shirts. The skirmishers used a bow, later the crossbow. The musketeers operated the pissides manuales, heavy muskets on stands, which probably did not do much damage, and were also not very popular.

Vessels

^{Die Nassaden waren zum großen Theile Bireinen; die leichteren Einreihenruderer. Die Autoren jener Zeit nennen sie Naves oder Nassadae duplices und simplices. Die Bordseiteu der Nassaden wurden vor der Schlacht mit Säcken, die eine Füllung von Moos, Haar und Wolle enthielten, geschützt. Einige hatten Eisenpanzer und Sp**n. Die Schiffe waren lang und scharf gebaut; der Bug lief in einen Wolfskopf aus. Alle führten ein bis zwei Masten mit Marsen zum Auslugen. Erlaubten es Wind und sonstige Umstände, so wurden Segel gebraucht. Wenn Mathias auch Thurmschiffe und Galeeren bauen ließ, so geschah dies augenscheinlich für Belagerungszwecke und zur etwaigen Absperrung des Stromes oder der Häfen, hauptsächlich aber zur Verwendung auf der unteren Donau und eventuell auch im Schwarzen Meere gegen die Türken; denn auf der offenen Donau zogen die Galeeren der Türken, welche 30 Jahre früher eingeführt worden waren, immer den Kürzeren gegen die schnellen und in ihrer Bauart den Stromverhältnissen sehr zweckmäßig angepassten Nassaden ; auch in den späteren Zeiten erwiesen sich die Galeeren wegen ihrer Schwerfälligkeit als unpraktisch auf dem offenen Strome. Geschütze wurden schon in der Zeit König Sigismunds eingeführt. Alle Nassaden führten solche auf dem Bug, die meisten hatten deren auch auf dem Heck.}

The majority of the Nassadas were biremes; the smaller part one-row-rowers. The authors of the time call them Naves or Nassadae duplices and simplices. The onboard sides of the Nassadas were protected from the battle with sacks containing moss, hair and wool. Some had iron armor and spur. The ships were long and sharp; the bow ran into a wolf's head. All led one or two masts with mastheads to look out. If wind and other circumstances allowed it, sails were needed. When Mathias also had tower ships and galleys built, this was apparently done for siege purposes and to possibly shut off the river or ports, but mainly for use on the Lower Danube and possibly also in the Black Sea against the Turks; for on the Upper Danube the galleys of the Turks, which had been introduced 30 years earlier, always lost against the Nassadas which were very appropriately adapted to the currents; In later times too, the galleys proved to be impractical on the open stream due to their clumsiness. Guns were introduced in the time of King Sigismund. All of the Nassadas had those on the bow, most of them had them on the stern.

Crew

^{Der Nassadist jener Zeiten war eine stahlharte Gestalt, seine Ausdauer eine geradezu staunenswerte. Tapferkeit und Todesverachtung zeichneten ihn aus. Auf dem Strome und dem Schiffe aufgewachsen, war sein Körper gegen jede Unbill des Wetters abgehärtet; früh schon lernte er die Handhabung des Riemens und des Steuers, sowie den Gebrauch der Waffen. Von Jugend auf an die Beobachtung der Strom- und Uferverhältnisse gewöhnt, deren Kenntnis sich überdies in den meisten Fällen von Vater auf Sohn vererbte, verstand er es sein Schiff unter allen Umständen mit Sicherheit zu führen. Er war gehorsam, pflichtgetreu und genügsam, wollte aber — stolz auf seinen Stand und dessen Privilegien — als Mann behandelt werden. Ebenso wie der Matrose zur See war auch er abergläubisch, glaubte an Himmelszeichen, an Wassernixen und Drachen, von welchen er Wundergeschichten zu erzählen wusste. Übrigens war er ein froher Geselle, wie aus den noch erhaltenen Liedern hervorgeht, liebte den Wein und war auch dem Weibe nicht abhold.}

The Nassadist of that time was a tough figure, his stamina astonishing. Bravery and contempt for death distinguished him. Growing up on the river and the ship, his body was hardened against all the hardships of the weather; early on he learned to use the belt and the steering, as well as the use of weapons. Accustomed to the observation of the current and bank conditions from an early age, the knowledge of which was in most cases passed on from father to son, he knew how to operate his ship with certainty under all circumstances. He was obedient, dutiful and frugal, but wanted to be treated as a man - proud of his status and its privileges. Like the sailor at sea, he too was superstitious, believed in signs of the sky, in water mermaids and dragons, of which he knew how to tell miraculous stories. Incidentally, he was a happy companion, as can be seen from the songs still preserved, loved the wine and was also not against women.