The last time I looked in to this, the consensus was that it was most likely a version of otherwise-extinct ancient Celtic.
Now that doesn’t mean that the Basques don’t have a potentially outsized Neanderthal genetic influence, but the odds of their language being so ancient as to pre-exist modern humans entirely is unlikely.
We can tell how much neanderthal ancestry someone has, more or less. Basque people have no more than others. Despite their odd language, they are much like other Europeans genetically: a similar mix of European hunter gatherers, Anatolian farmers and the bronze age invaders which we believe brought the IE languages to Europe.
This is what vibe-commenting from memory gets me.
This article talks about the Basque language from before contact with the Romans, 5-1 centuries BCE. It also references a "pre-proto-basque" language, that would have been the one before the Celtic invasion of Iberia (see https://en.wikipedia.org/wiki/Celtiberians).
The rabbit hole kind of ends there, as not much linguistic artifacts or history remain from BC unfortunately. But one can imagine the Basque society would live in relative isolation for a long time before that.