Hi OP, I'm going to guess that you're asking about pre-history here, rather than about hunters/gathers that have existed since (and still do). In that case, it may be worth x-posting to our sister sub, /r/AskAnthropology, since, while there are anthropologists here, there are more there, and more who specialize in earlier time periods.

This is an incredibly difficult question to answer, but for a part of it, you would study some historical linguistics. To be frank, it is impossible for us to know for sure about proto-languages. While we can trace certain common words between languages within the same family (like taking Indo-European quinque, panca, penta, and others to find a common root, penk^w e) going back as far as prehistory requires nothing short of a time machine. Taking a glance at rich, long-lived linguistic communities like West Africa and Papua New Guinea, we can see that in tribal settings with established territories, linguistic diversity flourishes. Nomadic groups in particular may share tongues of a common family, but develop wildly different pronunciations, meanings, and vocabulary from relative languages.

There was a flood of studies in the forties concerned about the so-called "first words", but much of it amounts to a "what-if" analysis, and the words inspected were mostly, if not all, Indo-European.

Anthropologists and linguists do tend to agree on the idea that language emerged once, and spread like wildfire. Because it was so long ago, we are unlikely to pinpoint where, but many look to Southern or Central Africa for the origin. Again, fairly impossible to designate.

tl;dr: Any study you will examine on prehistorical linguistics will be straddled with a hefty grain of salt clause.