Related question: Just how "Aramaic" would Babylonia have been at this time. I have read enough Mesopotamian history to understand that there were various waves of nomads related to the Arameans from the west who settled in Babylonia in the prior centuries, but I haven't really gotten a sense of how impactful their arrival was to the culture of this area.
During the Neo-Assyrian Empire, Aramaic was increasingly used as the lingua franca throughout the Near East. Its simple alphabetic script made it more accessible and provided a common language through which disparate and displaced populations could communicate. This was the case in Babylonia as well as in Palestine where Judaean populations would have used the language in everyday life. Hebrew continued to be used, though largely for liturgical purposes. As for Babylonia itself, it was a cultural melting pot where one could find communities of people from throughout the Near East, including North Syria, the homeland of the Aramaeans. I recommend this collection of essays for further reading.
From the time of the Neo-Assyrian empire (roughly the 9th century BCE), Aramaic takes over as the main spoken language in Mesopotamia as a whole. Akkadian is still used as an elite language - possibly spoken, but mostly restricted to writing and rituals - as late as the 1st century AD.
See Huehnergard, A Grammar of Akkadian, Eisenbrauns (2011), for a good (and brief) overview of the various forms of Akkadian throughout the millennia.