Italian Neo-realism predates French New Wave.
Edit: I had originally prefaced this with "I'm not a scholar" but forgot to readd it - the responses below invalidate most of what I said, but leaving the comment so their responses make sense - though I still stand by the lesser version of what I said - that Godard had a bigger impact on American cinema than pre-Fellini Italian Neorealism, and that parts of italian film influenced Godard.
It was actually the previous generation of French directors that they were all reacting against - who were making somewhat pretentious literary adaptation films. You can read about it in Truffaut's 1954 essay 'A certain tendency in the french cinema'.
Neorealism was actually a fairly minor influence - as none of the new wave directors were social realists. But older, maverick French directors like Bresson and Melville were much more influential, as it was their freewheeling sensibility which united all the new wave directors and sent them hurtling off in different directions.
40s: Italian Neo-realism
50s: Hollywood Golden Era
60s: French new wave
70s: New Hollywood
Somehow every wave greatly inspired the next one.
Me, personally, I’d also put the 1960s-1970s wuxia films from Hong Kong in a list of “stuff that has changed movie history for good”.