The sound track of "Le Souffle au coeur" is mostly Charlie Parker's tune.
No need to bring this film to listen to Bird.
Louis Malle and Vadim are not considered as "Nouvelle vague" director.
But their films as open the way to the "Nouvelle Vague" directors.
Jean-Luc Godard liked a lot "Et Dieu créa la femme".
"Détective", a Godard film from 1985 with Johnny Hallyday, Jean-Pierre Leaud, Nathalie Baye & Claude Brasseur, he features some Ornette Coleman Music.
In the press conference in Cannes that Year, He indicates than he chooses OC to be in the soundtrack because "the critic spoke about him as they spoke about me, saying: what he was doing before is far better than what he is doing now". The assistance laught a lot but I think he was dead serious when making this statement.
As the "Nouvelle vague" is concern, and at the exception of Jacques Rosier ("Adieu Philippine", music by Maxime Saury) only second rate "Nouvelle vague" directors use jazz as soundtrack like Edouard Molinaro in "Des Femmes Disparaissent" (Art Blakey & The Jazz Messenger).
Vadim as also uses Art Blakey for "Les Liaisons Dangereuses".
It's Jean-Pierre Melville, who was considered as the "grand frère" of the "Nouvelle vague" who tell to Godard to take Martial Solal for the score of "A Bout de souffle".
It was also Melville who recommend Martial Solal for write the soundtrack of "Les Parents terribles".
Melville, himself as always use "jazz ambiance" in most of his film noir but never hired a real jazz musician to write a score for him, except, again, Martial Solal partly for "Deux hommes dans Manhattan" (1958) and, before that, Bernard Peiffer (with guitare solo by Sacha Distel) for "Quand tu liras cette lettre" (1953). Eddie Barclay as also write the music of "Bob le flambeur" but the main was much on the side of the "international variety" than real jazz. In "Le Doulos" the "piano bar music" who give his mood to the last spoken scene of the movie was writte (but not play) by Jacques Loussier.
If you want find jazz in french film, you must look for the second or third generation of post "Nouvelle vague" directors such as Jean-Louis Comolli or more straight one as Bertrand Tavernier or Claude Sautet.