If this is the case, then the first language would have probably rapidly spread between the early populations in africa – if, indeed, there were multiple distinct populations at the time – and new (homo sapiens–originated) languages would have only arisen if the chain of speakers were somehow broken and children raised without access to an existing language.
With that being said, there are some cases where something similar has happened. Most obviously are sign languages, which are believed to have had multiple origins, since it is possible for deaf people to be born among non-signing populations. Something similar in the case of spoken languages is the phenomenon of pidgins evolving into creoles mentions elsewhere in the comments, although they still retain at least some aspect of one of their parent languages. The extent to which the latter count as proto-languages is questionable.