Forget French musicals. There aren't enough of them and they probably don't exist in French in a form that is easily available.
Irma La Douce is one.
Umbrellas of Cherbourg (LeGrand) is a French film that was all sung but the 'lyrics' are unusual. The dialogue is sung, but it is only dialogue not lyrics in any conventional sense of the term. English lyrics were fashioned for the two main themes, but they were written by the Bergmans I think. They are nowhere near literal translations. The songs are "I Will Wait For You" (which was a 1964 Oscar nominee) and "Watch What Happens" which I think is the better song of the two.
Listen to Edith Piaf recordings or Charles Aznavour and then find translations so you know what the songs are about.
"If my life weren't funny, it would just be true. And that would be unacceptable."
--Carrie Fisher