Is it possible to estimate the distribution of Jews in the variety of cities they settled in the aftermath of expulsion from Palestine by Adrian in 135 AD and in the following few centuries of the late antiquity? In particular, are there any source on what were the numbers of Jewish communities in Alexandria, Carthage, Rome, and other large cities of the Empire? I would like to learn how the numbers in these communities changed between 2nd and 5th centuries AD, including their percentage within population, were there any large scale migrations not immediately after 135 AD, etc.
Here's a partial answer covering just Alexandria (though also see comment about Carthage below).
The most significant event in the history of the Hellenistic Jewish community of Alexandria was not the end of the Bar Kokhba revolt in 135CE, but the suppression of the Rebellion of the Exile (Kitos War) in 117CE. Theodor Mommsen estimates that in the first century CE, there were approximately 200,000 Jews in Alexandria out of a total population of 500,000; this estimate is likely to be from Philo, who also claimed two of the five city districts were majority Jewish.
Following anti-Jewish riots in 38CE, the Jewish community was concentrated into one of the five districts. The suppression of the Rebellion of the Exile in 117CE resulted in the near extinction of the community in Alexandria. The fate of the surviving Jews is unknown: some may have converted to Christianity or Gnosticism.
The Jewish community reappears in the 4th century in a more Hebraized form (reflecting a strong Palestinian influence). In 415CE the community is expelled by St. Cyril though it appears again soon after (the expulsion may have been partly symbolic, affecting community leaders more than others). When the Arabs arrived in 642 CE, Amr ibn al-As reported 40,000 Jews paying head tax (jizyyah).
Taking the numbers above at face value would therefore suggest the following approximate history: ~40% up to 117 CE, very little for two hundred years, then growing to ~10% by 600 CE. I don't know of any specific large scale migrations during this period, with the possible exception of the expulsion in 415.
For some more details see Jewish Survival in Late Antique Alexandria by Guy Stroumsa.
The Jewish population of Carthage was both newer and smaller than that of Alexandria: substatial settlement dates to the Roman Empire and the community was primarily Latin speaking. According to the Jewish Virtual Library, the community grew after both 117CE and 135CE due to migrations, and shrunk in the 6th century due to religious persecution under Justinian. A Jewish necropolis providing space for as many as 1500 burials was dated to the third century; the size of the necropolis suggests a relatively populous community, but I've not seen any attempts to estimate the Jewish population. Sadly, there are significantly fewer physical remains from Carthage and its vicinity than from Egypt or Cyrenaica. For a summary of what is known about the community see The Cambridge History of Judaism (Vol 4, pp68-75).