In 1764 BCE (or about 1930 BCE adjusting for the missing years (Wikipedia link), everyone (perhaps under Nimrod's command?) had built the Migdal Bavel, and that Hashem caused a dispersion and confusion of their unified language due to this (Tower of Babel) (Genesis 11:1-9):
Now the entire earth was of one language and uniform words. And it came to pass when they traveled from the east, that they found a valley in the land of Shinar and settled there.... And they said, "Come, let us build ourselves a city and a tower with its top in the heavens, and let us make ourselves a name, lest we be scattered upon the face of the entire earth." ... And the Lord said, "Lo! [they are] one people, and they all have one language, and this is what they have commenced to do.... Come, let us descend and confuse their language, so that one will not understand the language of his companion." And the Lord scattered them from there upon the face of the entire earth, and they ceased building the city. Therefore, He named it Babel, for there the Lord confused the language of the entire earth, and from there the Lord scattered them upon the face of the entire earth.
On this, Rashi says that they all spoke Lashon Hakodesh (presumably Hebrew) and that it was a gathering of all the different peoples ("One nation to another nation, Mizraim to Cush; and Cush to Put; and Put to Canaan"). As it was less than 400 years after the Mabul (Noah's Flood), I doubt there was much time for language development anyway.
What exactly does this mean? Put more directly, what languages did people speak before and after the dispersion of Migdal Bavel?
Note that it seems from archaeology (Wikipedia link) that there were other written languages continually in use both before and after that time, perhaps most famous of which is Egyptian hieroglyphics (yet interestingly no record of Hebrew or Aramaic writing until several centuries later). For all I know the archaeology could be partially or entirely incorrect, but it is worth keeping in mind in context of this question. But beyond the simple reading of the verses, what do the actual Torah sources say about what languages were in use before and after Migdal Bavel and what effect the dispersion had on language?
Note also that this question has some relevancy to mine (though I couldn't find one specifically about the dispersion's effect on language).