My question specifically what would happen to France in the aftermath of losing World War I. There are going to be two immediate options, either the Republic survives or it doesn't.
The Third Republic had survived over 45 years at this point, so maybe it continues, since no one can agree on an alternative.
Some kind of French Kapp Putsch is definitely a possibility. I think the eventual French form of government is going to be a response to why the French lost the war. The French mutinies of WWI loom large in my mind, because I think that the support for wartime Army leaders is going to be very, very low (i.e. no French Hindenburg), and this lack of support may translate into large numbers of veterans who drift in a radically leftward direction (I'm thinking the Second Commune).