Two for the Road – This is the first mainstream American film I can think of to have a nonlinear plot. Fascinating in that regard.
What about Citizen Kane (1941)? Also, isn’t that a British film?
It’s directed by Stanley Donen and came through the studio system (Fox), so if it’s not American, it’s still Hollywood. And Citizen Kane was kind of linear, really, just going from flashback to flashback from person to person. Not to deny that film its due. Two for the Road is more avant garde in regards to structure, the timeline for the movie goes back and forth with little actual reason, with different periods merging together seemingly. I’d compare it to Eternal Sunshine of a Spotless Mind, in that it feels like you’re exploring this couple’s memories.
I’m just going off Wikipedia and IMDB, which both say it’s from the UK. I’ve never seen it. I get what you mean about Kane, in that it’s technically framed as flashbacks (and newsreels and such), but I think it’s inaccurate to call that linear.