The French nuclear umbrella being extended to other EU states is still only a suggestion (though even as just a suggestion it is a very strong political message). What that would mean for French nuclear doctrine in practice is yet unclear. Would France nuke Russian troops/infrastructure/cities and risk all-out nuclear war to protect Romania? Moldova?
Funnily enough the entire reason why France has an independent Nuclear program at all is De Gaulle did not trust the US to risk New York being glassed in order to save Paris from a Russian invasion. Which was probably correct. Unfortunately, the reverse logic also applies and France will not risk Paris being glassed to save Toronto. The only reason why a sovereign European nuclear umbrella makes sense on paper is that an attack on any EU member state hits close enough to home as to arguably be existentially threatening, unlike a war an ocean away.
Unless you meant France selling nukes, but that would violate every nonproliferation treaty out there and just be a complete mess that even with a sane US administration would lead to a complete diplomatic meltdown. The current suicide cult at the helm would probably actually start a nuclear war for less than that.
It’s French but has a weird legal status where the citizens are French and therefore have an EU passport and vote in EU elections, but it is not part of the EU and Schengen from what I understand.
Anyway technicalities aside, France generally doesn’t care much for its overseas territories. Quality of living varies wildly from territory to territory and there’s still ongoing mid 20th century style colonial oppression in places like Nouvelle-Calédonie.
So while a Falklands type situation is quite possible if anyone tries to invade an overseas French territory, it’s doubtful that France would risk actual nuclear war over one. Especially Saint Pierre and Miquelon which does not have a strategic military value as far as I can tell, unlike other territories which serve as force projection multipliers and have naval bases, especially for the Charles de Gaulle aircraft carrier and the nuclear deterrence submarines. Without those France wouldn’t be able to operate in the Pacific theater.