This was a BIG issue for me. I worked on this issue constantly in the latter part of 8th grade and the first half of 9th grade, until I gave up. I wanted so much to make "history" fit into the WT chronology. According to the Society, the Flood was in 2370 BC and Abraham was born in Ur in 2018 BC. It wasn't clear when the tower of Babel would have been built, but I used the reference to the "division" of the earth in the days of Peleg as the main clue. Peleg was born in 2269 BC, and since he was named when he was born, the division of peoples at Babel would have occurred when he was born. Plus, 2269 BC, being about 100 years after the Flood, would give barely enough time for people to be born to populate the cities of Nimrod's kingdom and to be scattered to populate the rest of the earth. It was also desirable to have a date as early as possible (unlike the date the Society suggests for the dispersion, 2189 BC) to give as much time as possible for Egyptian history and other peoples outside of Mesopotamia. At least with Sumerian history, I would have the advantage of being able to use that first century after the Flood.
So I started Sumerian history about 30 years after the Flood, to give enough time for at least a few families to build cities in the land. Then I arbitrarily omitted about the first half of the first dynasty of each city as "mythical kings", and then shortened the reigns of the other kings to whatever length to make them fit. I shortened the reigns of the kings of the second and third dynasties as well on a similar basis. I also tried to maintain the synchronisms between the dynasties (i.e. making Gilgamesh of Uruk and Agga of Kish reign at the same time). Now I had a definite idea who Nimrod was. I thought when he was young he was known as Gilgamesh and built Uruk (so I omitted all kings who reigned before Gilgamesh, except for his father), and when he grew older he moved out, built Babel as his religious city and Akkad as his political city and became known as Sargon of Akkad. This meant that all early Sumerian dynasties had to precede Sargon's reign, and that Sargon's reign would have started by 2269 BC, if not sometime before. That also was barely enough time to squeeze in the Akkadian dynasty of his descendents and the Third Dynasty of Ur, to end shortly after Abraham was born so the city could still exist in his time. With those parameters, I had to reduce the lengths of Sumerian reigns down to like 1 year, 3 years, a really long reign would be 5 years, many were less than a year. Finally, I had a workable chronology, but then I looked at it and realized how implausible it was. Just a few decades after the Flood there were enough people born (the Bible mentions only 70), to populate several cities with competing dynasties, and then each city had very quick dynasties lasting just a decade, with many kings ruling for mere months, and then after the dispersion of Babel, the Egyptians had to hurry over real quick to Egypt so they could start all those Pre-Dynastic and Old Kingdom dynasties as fast as possible, because I didn't have much time to squeeze those in either, and then make the Chinese hurry fast over to China to start the early Chinese dynasties, etc. And to make the Egyptian dynasties work, I had to also shorten their reigns (I didn't have the luxury of regarding them as "mythical" since most are known historically), but this posed a problem because I needed enough time for each king to build their own pyramid, etc. But the death knell for my attempt at harmonizing the Society's chronology with history turned out to be non-chronological things I didn't even think of. I read about the royal tombs of Ur and the tombs of King Meskalamdug of the first dynasty of Ur and his wife Queen Pu-abi. Well, there were about 660 graves in all dating to the Early Dynastic period (i.e. before Sargon), and the royal tombs had the bodies of dozens of slaves executed so they would die with their king. Well, that wouldn't work at all! Genesis indicates that people generally lived for centuries after the Flood, so there shouldn't be any graves at all (or very few), and I had been assuming so far that cities in the first dynasties after the Flood had less than a 100 people to start out with....so why would there by so many dead people? And if they're trying to repopulate the world and populations were still fairly small, how in the world would they have the luxury of killing off dozens of people just so they would die with their king or queen?
So I tried to change the basic parameters of my harmonizations, giving much more time for the early Sumerian dynasties. But that just took away critical time from the Egyptian dynasties, which were already some 800 years out of synch with secular chronology and I had to squeeze it down further (I tried to crunch down the First Intermediate Period and the Middle Kingdom). Plus it just seemed implausible that the sons of Mizraim would race quickly over to Egypt, build cities immediately, so that I would have as much time as possible to squeeze in all the kings and give them reigns as long as I could manage.
Eventually I just gave up, and realized that the chronology could not be harmonized with history. Since I was not allowed to think this, I put the issue out of my mind for a few years.
