That would be silly, given the amount of time they've been afflicted with the curse. Gutterspeak antedated the Forsaken—it was the "low" vulgar speech of criminals, possibly a linguistic substrate or vernacular. Might have been influenced by the tongues of the savage races (Gnolls, Kobolds, etc). The Forsaken adopted Gutterspeak as their official language as a measure of forging a new, distinct, and separate culture vis-a-vis the Humans with whom they no longer felt any great camaraderie. One might compare it to the real-world revival of Irish.
For what it's worth, It is noted in RPG supplements that feral Worgen do have their own language. Maybe the curse imparts knowledge of this language—it wouldn't be too far outside the realm of fantasy cliché and would possibly explain how Crowley and Ivar Bloodfang communicate. But the fact that it's called "Gilnean" and not "Worgen" or "Wolfspeak" or somesuch seems to imply that the language is particular to the Humans of Gilneas, both before and after their transformation.