I think you'd only expect to find a significant difference in the rates of autism between regions where interbreeding with Neanderthals occurred, and regions where it didn't.
That seems highly dubious. The global human population isn't that well connected, or at least wasn't until recently. And single genes aren't that easy to isolate, so if they could travel that quickly we would also see all kinds of more visible traits moving around, but instead for most people we can get a decent idea of their genetic heritage looking at their face.
All people on Earth are descended from Neanderthals, and East Asians have the highest ancestry. The idea that only Europeans are related to them is outdated.
You would also expect autism to be nearly completely absent in Africa, East Asia, and native Oceania and North and South American populations.
I’m not an expert, but I don’t believe any of that is true.
It’s a fun “theory” but it doesn’t survive even casual analysis.