I’m not quite the right person to ask about this; serious interest in the origins of language resumed after I studied linguistics.
But think about it. Why do words have multiple meanings?
We differentiate polysemy and homophony: multiple related meanings, and multiple unrelated meanings.
Why is there polysemy? Because words get applied to different contexts, by analogy and metaphor and metonymy.
When would polysemy have started? The minute humans became cognitively capable of analogy and metaphor. And that capacity may well have predated language.
Why is there homophony? Look at Marc Ettlinger’s answer to What is the reason for the existence of polysemous words in a language? (Even if he’s addressing homophony rather than polysemy.) Accidental convergence, borrowing, neologisms.
When would homophony have started? The minute there was more than zero neighbouring languages to borrow from, and the minute sound change started, and the minute people started making up new words. And that would have been not long after humans started using language.