I can only really answer this in relation to the Caribbean, but a good portion of it was the evolution of the various pidgin and creole languages (the biggest of which is probably French Creole, spoken by about 8 million in Haiti alone). Pidgin is basically a shorthand expression that doesn't really exist, whereas creole is a fully formed language that's just grammatically and syntactically simpler than the base language (usually a colonial one such as English, French, Spanish, or Portuguese).

As for the historical evolution, obviously slave owners in the Caribbean didn't speak African tongues, yet still expected understanding from their subjects. From that, Creole(s) evolved over time to allow communication, eventually becoming a distinct language that was (and still is) viewed as inferior to the original.

To answer your question, there wasn't any real set time that slaves stopped speaking their native languages. Rather, it's more similar to children of immigrants today, where after a few generations the skill passes and the descendants can understand the language but not speak it, eventually losing the skill. It's important to remember that the slave trade was a process over multiple generations, not within a short time span, and as such much would have changed. This is also combined with the fact that disuse of a language can lead to forgetting it (including the native tongue), as well as likely disdain from slave owners for use of African languages, making the various creoles the more commonly used languages by the slaves.

To clarify one last thing in regard to my use of Creole(s); I usually refer to Creole as the Haitian language, but it's by far not the only Creole to exist and there's currently some linguistic debate over the interrelation of Creole languages that's tangential to this conversation but still semi important.

I would also recommend asking a professional linguist or two (there may well be some in this sub), especially one that specializes in a pidgin or creole language, as they might be able to help outline how the exact evolution of the language occurred and why in better detail. I'm approaching your question more from the perspective of heavy background in the surrounding context, as well as a speaker of all the relevant colonial languages.

EDIT: Forgot to mention that because so many creoles and pidgins are more informal and see significant more verbal than written use, it's somewhat hard to pin an exact time on their evolution until the last century or so.