What was the device that general Birdwood was using in the "Gallipoli" TV series?

One possible explanation is the use of "sound ranging".

During World War I, a team of scientists developed a technique called sound ranging which worked using a system of microphones and a device called a string galvanometer.

The purpose of the equipment was to determine where enemy guns were positioned by analysing the length of time it took sound impulses from the firing of guns to arrive at the allied front.

The equipment would have had six "microphones", the signals from which were recorded simultaneously on the film recorder.

Another, possibly more applicable to masses of troops disembarking from ships and exposing more of the ships' hulls as the ships got lighter, at the start of the Suvla Bay campaign and thus creating a larger disturbance in that location of the Earth's magnetic field.

"As a matter of actual personal experience," said Dr. Tesla. "it became a fact that the small iron-hull steam mail-packets (ships) plying up and down the river Seine at a distance of 3 miles would distinctly affect the galvanometer!"

... this magnetic method of locating or indicating the presence of an iron or steel mass might prove very practical ...

The device in the picture looks like a signals galvanometer. Such devices were used to monitor signals traffic, by measuring changes in the localized magnetic field produced by spark Morse code transmitters, that would have increased as the campaign commenced.

I suspect that they were dramatizing the use of the galvanometer in "cable" signaling. This differed from Morse "landline" signaling in that undersea cables have inherent capacity effects from the core connecting wire and the outside (sea water), therefore Morse cannot be used on undersea cables.

Undersea circuits use a galvanometer to receive with, and is received by reading positive or negative (needle swings one way for a "positive" energized cable, to signal a "dot") and the reverse is true for energizing the cable by reversing the polarity, causing the needle to swing in the other direction (thus making a "dash"). The sea was used as the "common" or second side of the circuit.

Later, siphon registers were used to imprint the wavy ("+" or "-") line onto a moving paper strip that was read similarly to the way Morse printer tape was read with dots and dashes. I hope this helps clear this up a bit.

–Robert Feeney, Telegraph Historian and Collector

Assuming that it is indeed a galvanometer, as LаngLаngС suggests (and I think they're right), then my suspicion is that it is indicating something to do with the telegraph signalling system from the landings.

There is a detailed appendix in the war diary of the GHQ Director of Signals (WO 95/4268, Ancestry). This sets out the signals work at Suvla; a party was sent ashore in the night at Nibrunesi Point, overlooking Suvla Bay, to bring ashore a cable. The idea was that they would then be able to relay signals between the landing divisions, the ships, and headquarters until those units could also be connected to the telegraph system.

The cable ship was in position at 0115, and the shore party signalled over the cable that it was connected at at 0209. They lost contact 0333-0408, but came back on the air at that point. Wireless sets did not arrive until 0515, and contact with the beaches was made at 0525. A cable down to the beaches was set up at 0550, wireless at 0615, and Morse signalling set up at 0623. GHQ noted:

Very few messages came through during the day - until 0848 nothing was recieved except SGs, and till then GHQ was kept informed of progress of landing by information from Signals.

I strongly suspect, but with the caveat that it's speculative, that this initial connection is the event that is being dramatised. The first confirmation that anyone is ashore comes from that telegraph line being connected, and presumably you would be able to easily hook another indicator into the line so that it will start flickering when there is a live circuit. That line runs through a headquarters where Birdwood is based, even though it's not formally reporting to him (he wasn't in command of the force landing at Suvla).

So my guess is that it is a galvanometer being used as a galvanometer, to indicate the telegraph line is live and thus that the shore signal station is now active.

Whether this actually happened, or if it was dramatised... I don't know. It seems a bit of an odd thing to make up, but equally it could have been an incident from elsewhere in the war that was added here for dramatic effect. I can't find any mention of it elsewhere. The Australian official history doesn't go into much detail about the landings, but on the afternoon of 7th August it notes:

Of the progress of the force disembarked at Suvla Birdwood had no information except that which his eyes afforded him. (p 646)

The official history doesn't seem to mention any specific notification of the landings overnight, and it is not in Birdwood's memoirs. I have not been able to track down anything from other eye-witnesses. Ian Hamilton (to whom the signals were all ultimately going) noted an "All landings successful" cable in the early morning and since then 'no direct news'. (memoirs, vol 2, p. 56).

It's technically a galvanometer. Available then in principle for quite some time.

They were the first instruments used to detect and measure small amounts of current.

It was used before and during the war as metal and gun detector, and perhaps even as a very sensitive seismometer.

If you want to get one you may find it under "Vintage Military Detector":

World War II military detector W.E.M. Co Ltd No2753 1942. This is a WW2 British dated 1942 Detector Q & I A.T.PIT Reads On The Dial Detector Q & I.A.T.P. (WAO275 ) W.E.M Co LTD No 2753 1942

— src: Vintage World War II Military Detector 1942 W.E.M.CO Ltd NO2753

Another listing decribes such a device as:

Antique Edison & Swan Q & I detector meter galvanometer […] for a larger view. Description: Brand: Edison & Swan Condition: Antique VTG Old Weight: 600 gr Antique Edison & Swan "Q and I" detector meter CA 1917, was made by Edison and Swan United Electric Light Co. Ltd An elementary galvanometer made for telegraph, telephone and other types of electrical maintenance workers to make rough battery checks, find faults, number wires, and other similar tests.

Or:

Edison and Swan "Q and I detector," – This British Q and I detector is an elementary galvanometer made for telegraph, telephone and other types of electrical maintenance workers to make rough battery checks, find faults, number wires, and other similar tests. It is not intended to measure absolute current or to be part of a working telegraph set. The "Q and I" stand for "Quantity and Intensity," 19th century electrical terms, which are misleading terms to describe the coils. The "Q" coil has a resistance of .2 ohms. The "I" coil has a resistance of 100 ohms. The "intensity" coil is used for localizing faults in lines using a series resistance. The "quantity" coil is used primarily for battery testing.

The uses for that were apparently quite widespread:

Edison and Swan Q and I detector

The company Edison and Swan was a joint venture company formed in 1883 between the American inventor Thomas Edison and the British inventor Joseph Swan. Both had clashed over the invention of the electric light bulb in 1878.
This instrument was apparently used by telegraph linesmen for checking batteries, finding faults, and identifying and numbering wires. It comprises two coils, the Quantity (Q) coil rated at 0.2 ohms and the Intensity (I) coil rated at 160 ohms.
The Q coil is used for testing batteries and the I coil for localising faults.
A current of 140 mA in the Q coil should produce a needle deflection of 30 degrees and 10 mA through the I coil should cause a deflection of 50 degrees.

This instrument carries the markings “Edison and Swan, No. 22386” is dated 1918 and comes with a leather carrying case.

— Computer Networking and Telecommunications Research: "Telegraphy: The birth of modern communications", University of Salford, undated.

These could be used as follows:

Historically, the first reference to gun detectors was in a patent filed by C. N. Clark on 12 April 1895 (Patent No. 541, 719).

Clark placed in a building a large air coil electromagnet connected to a galvanometer relay. He claimed that when persons came near to or walked away from the coil, the needle of the galvanometer would be deflected.

In his patent application, Clark stated,

"It is further understood, that any stationary iron in the magnetic field would not produce a deflection. But whenever a person having metal upon his body moves into the magnetic field or away therefrom, the galvanometer is actuated. On the other hand, a piece of soft iron may be suspended or arranged near the coil, so that when a person not having any iron on his body approaches or moves from the magnetic field, the jarring of the floor or other part of thie structure causes the soft iron to vibrate and disturb the magnetic field of the coil and conse- quently, causes a deflection of the galvanometer." The idea, then, of gun detection based on magnetic principles is at least 77 years old.

— James H. Henry, Institute for Defense Analyses (IDA): "Theory And Application Of Magnetic Weapon Detectors", Protection of Public Figures, Symposium Proceedings, U.S.Army Mobility Equipment Research, And Development Center, Fort Belvior, Virginia, May 16–18, 1972. PDF

So it is a device that comes from the field of electricity and was indeed widely used in telegraphy. The early patent granted for a military application of it leads me to not being able to answer the subquestion: How that device lead to astonishment by the other soldiers present?

As Andrew's answer notes, the actual monitoring of the landing progress apparently was not done with 'advanced technology'. That leads to the assumption that this slightly vague detail in the TV series is not entirely historically accurate to begin with. Which leaves us with a lot of speculation over how such a galvanometer could be used for such a task, of which there a quite a few possibilities imaginable (some found in the recommendable other answers on this page). And further speculation as to why the writers included the device.

For the latter, a working theory might revolve around Lawrence Bragg, an Australian, who won in 1915 a Nobel Prize and in the same year developed the technology of sound ranging (SR) for artillery fire that involved a string galvanometer, shortly after his brother was killed at Gallipoli.

All these points towards a connection with the theme of the show. However, as was correctly noted in comments already, the actual real SR technique was not suitable at all for the conditions found at the landing. A setup from the time looked more like this:

— src: UK Military Survey Historic Archive

More on Bragg and SR:

— Richard Daniel Costley Jr.: "Battle eld Acoustics in the First World War: Artillery Location", Acoustics Today, Vol 16, No 2, 2020. doi
— William Van der Kloot: "Lawrence Bragg's Role in the Development of Sound-Ranging in World War I", Notes and Records of the Royal Society of London, Vol. 59, No. 3, 2005, pp273–284.

Altogether this makes it quite in my eyes quite plausible that the writers connected quite a few dots and conflated the info on 'Australians, new invention, galvanometer' into this little device as 'newest tech' without much needed and/or detailed explanation on screen?