Unfortunately, the research I had done which was neatly laid out was on a HDD that died on me last year. From what I can remember the historical list of kings are:
Nebuchadnezzar (ruled 43 years)
Evil-merodach (ruled 2 years)
Neriglissar (ruled 4 years)
Labashi-Marduk (ruled 9 months)
Nabonidus (ruled 17 years)
Counting back from 539 this leads us to Nebuchadnezzar starting his rule in 606.
The following quotes have the WT admitting that Nabonidus ruled 17 years(the ruler they put into question as there being two Nabonidus'):
"Other investigators say this: "The Nabunaid Chronicle . . . states that Sippar fell to Persian forces VII/14/17 [Footnotes]"VII/14/17": The 7th Hebrew month Tishri, 14th day, 17th year of Nabonidus' reign . (Oct. 10, 539), that Babylon fell VII/16/17 (Oct. 12), and that Cyrus entered Babylon VIII/3/17 (Oct. 29). This fixes the end of Nabunaid's reign and the beginning of the reign of Cyrus." Watchtower 1968 August 15 p. 491
"Last supreme monarch of the Babylonian Empire; father of Belshazzar. On the basis of cuneiform texts he is believed to have ruled some 17 years (556-539 B.C.E.). He was given to literature, art, and religion. Insight on the Scriptures, Volume 2 p.457 Nabonidus
The WT has to follow up with their faulty timeline by saying there were actually two Nabonidus'. There was one that ruled over a city 8 years into Nebuchadnezzars reign (and eventually was ruler of Babylon). WT timeline puts it at 616 to fit their timeline meaning he would have been way too old to rule a city back in 616 and still be ruling at 539 so they had to invent another Nabonidus for which there is no historical record whatsoever. They know that Nabonidus ruled directly after Labashi-Marduk and that there was a Nabonidus ruling when Babylon fell. So that must mean there are TWO of them since he would have been too old to have been ruling Babylon when it fell. To quote them:
"Cuneiform tablets of the eighth year of Nebuchadnezzar (Nisan 617-Nisan 616 B.C.E.) list a certain Nabu-na´id as the one "who is over the city," and some historians believe this is the same Nabonidus who later became king. However, this would mean that Nabonidus was a very young man when placed in such administrative position and would make him extremely aged at the fall of Babylon, some 77 years later (539 B.C.E.)." Insight on the Scriptures, Volume 2 p.457 Nabonidus"
Of course there is no need for a second Nabonidus if you go by ACTUAL history. They attribute 36 years to the two kings... without any shred of historical evidence to back it up except that it is the only way their absurd timeline fits.