Is the story of Peter Niers brutal execution authentic?

There is actually a contemporary account of Niers's crimes and execution. It can be found in a short pamphlet published shorty after his death:

"Warhafftige Newe Zeittungen / Erschröcklich unnd Erbärmlich / so alle kurtzlich in disem 1581. Jar geschehen seind / und auff das kürtzeste verfasset", Heidelberg: Jacob Müller 1582 (scan online)

My translation: "Truthful news reports, horrible and pitiful, about [events] that happened recently in this 1581^st year, and written up in the briefest way."

Towards the end one finds:

Auch hett ich schier eine schreckliche zeitung dahinden gelassen / Von dem grausamen unnd schröcklichen Mörder / Peter Niers genannt / der zu Neumarck / fünff Meil von Nürnberg gelegen / fünfhundert und vier und viertzig Mördt gethan / hat vier und zwentzig schwangere Weiber umbgebracht / wie er auch entlich hernach gericht ist worden / Drey tag hat man ihn gepeiniget / zum ersten Riemen aus seinem Leibe geschnitten / und heiß Oel drein gegossen / Den andern tag die Solen an Füssen angezündet / Am dritten tag hat man ihn hinauß geführet / und im zwen und viertzig stöß mit dem Rad geben / darnach geviertheilt / unnd auff vier Strassen gehencket / Also laßt Gott das ubel nicht ungestrafft / das gut nicht unbelonet / Datum den 16. Septembris / Anno 1581.

My translation: "Finally I left one piece of horrible news to the end, of the barbarous and terrible murderer called Peter Niers, who committed 544 murders in Neumarck *, five miles from Nuremberg, and killed 24 pregnant women. This is how he was executed in the end. He was tortured for three days. On the first they cut out strips of skin from his body and poured hot oil into [the wounds]. On the second day they burned the soles of his feet. On the third day they led him out [of the town] and gave him 42 blows with the wheel. Thereafter he was quartered and hung along four roads. This is how god does not let evil [deeds] go unpunished and good [deeds] unrewarded. Date: September 16, 1581." [* Neumarkt in der Oberpfalz]

Given the heinous nature of Niers's (alleged) crimes, the methods of torture and execution described do not seem particular unusual for the time and thus seem entirely plausible.

The Deutsches Rechtswörterbuch (Dictionary of Historical German Legal Terms), provides three citations from legal codes from the south of Germany and Austria for Riemenschneiden (literally: strap cutting; cutting away strips of skin) from which is clear that this method of torture was added as an "enhancement" to the ordinary punishment due in cases of serious crimes.

As for torture by hot oil and burning, similar treatment was meted out to Robert-François Damiens in 1757 for the attempted but unsuccessful assassination of the French king. According to Wikipedia:

He was then tortured with red-hot pincers; the hand with which he had held the knife during the attempted assassination was burned using sulphur; molten wax, molten lead, and boiling oil were poured into his wounds.

Various European legal codes prescribed breaking with (or on) the wheel for murder and other serious crimes. This method of torture was designed to break the bones of a delinquent without killing them. The practice was only abolished in various German-speaking regions during the 19th century. Wikipedia describes it as follows:

Therefore, the most common form would start with breaking the leg bones. To this end, the executioner dropped the execution wheel on the shinbones of the convicted person and then worked his way up to the arms.

The manner of quartering described in Müller's pamphlet seems consistent with that prescribed in the Constitutio Criminalis Carolina (the legal code of emperor Charles V) of 1532 which applied to the entire Holy Roman Empire to which the region around Nuremberg also belonged. As cited by Wikipedia:

Zu der Viertheylung: Durch seinen gantzen Leib zu vier stücken zu schnitten und zerhawen, und also zum todt gestrafft werden soll, und sollen solche viertheyl auff gemeyne vier wegstrassen offentlich gehangen und gesteckt werden

My translation: "Concerning quartering: [He] is to be cut through his entire body and hacked apart into four pieces, and thus punished by death, and the quarters shall be hanged and staked publicly at four commonly-used roads."

A note on my translations: I have translated rather freely, and my attempts may not capture every nuance of the original historical German, which is about as similar to modern German as Shakespeare's English is to modern variants. I have enclosed as few clarifying additions in square brackets. Suggestions for corrections and improvements are welcome.