Rashi in parshas Noach 10 11, seen here quotes the Medrash Rabba: From that land: Since Asshur saw his sons obeying Nimrod and rebelling against the Omnipresent by building the tower, he departed from their midst. — [from Gen. Rabbah 37:4]
It seems quite apparent from from the repeated use of the word 'sham', there, in chapter 11 vs. 7-9 that the creation of new languages took place only in Babylonia.
So shouldn't the Assyrians have continued speaking Lashon Hakodesh being that they weren't in Babylonia when Hashem mixed up the languages?