Over about 2500 years, the Middle Eastern farmers occupied almost all of Europe. That migration seem to have a single origin: genetically they seem quite similar, even as far as north as southern Sweden.
The indigenous foragers didn’t disappear instantly, and some seem to have survived amidst the dominant farmers for a long time, without much gene flow. This pattern looks similar to what we’ve seen with Pygmies and Negritos. In these contemporary examples, the encapsulated foragers have lost their original language. Probably those persisting European hunter-gathers also adopted the language of the surrounding farmers.
If this picture is correct, it seems likely that almost all of Europe, circa 4000 BC, spoke related languages, languages descended from a language that originated in the Fertile Crescent. The language map probably looked something the Bantu languages in Africa.
And then things went bad for the EEF peoples. There is only one surviving language that seems likely to be member of that old radiation – Basque. There were other languages that were probably in that family and lasted for quite a while: Iberian, Nuragic, probably Tartessian, maybe Etruscan, possibly Minoan and Eteocypriot. The evidence we have on those other languages generally stems from the Classical era: by that time, these languages had had maybe five or six thousand years to diverge.
This means that Theo Vennemann is probably right about a Vasconic substratum, only it should be considerably more extensive than he suggests. The population in question originated near the heartland of Middle Eastern agriculture (First Farmers would be a good ethnonym), rather than spreading from a glacial refugium like Spain after the LGM.
How to prove this? Sounds tough. We may need another Rosetta stone.