> if there were several populations of humans developing speech independently of each other, and at different times

We already know of instances of societal-cultural upheaval or even trauma, so severe, that the language undergoes unrecognizable transformation in the form of pidgins. A group of young people from scattered tribes after some disaster, who speak no common language, and who are barely lingual and "re-invent language" for the next generation, seems quite possible over the last several tens of thousands of years, perhaps in multiple times and places.

The similarity of grammar, in unrelated creoles, descended from such pidgin languages, is one of those great tantalizing mysteries in linguistics, and in my view, one that suggests we might just be wired to make a language if we didn't already have one, as do things like the apparent spontaneous emergence of sign languages in deaf communities. The spanner in the workers for that theory is that, of course, those hypothesized people who didn't speak related languages still grew up in lingual communities and were exposed to the idea of language from birth (even the deaf ones, really). Perhaps such events are just a remixing of what already was culturally common, and common to all humans. Indeed, perhaps no community where the link of language transmission has been completely broken, could even survive.