The answer to your question lies mainly in the fact that there was never any "Celtic" ethnic group spanning throughout Western Europe, the term is also used to denote a linguistic group or a material culture. All "Celtic" peoples are not even necessarily culturally linked let alone ethnically related. You should check out these posts:

• Who were the ancient "Celts" that lived throughout Europe during the time of the Roman Empire? Why is it today anything "Celtic" usually always refers to only Ireland? by /u/Daeres

• What caused the Celtic expansions across Europe, and then the later Germanic expansions? Were there any shared causes? by /u/Miles_Sine_Castrum

• If the Celts didn't exist, what ethnicities existed in the areas that spoke Celtic languages? by /u/innisfallen

These all come from the FAQ on Celtic/Gaelic Cultures. I hope this helps you!

As /u/Dinocrocodile mentions, there's a bit of a disconnect between the popular usage of Celtic (evoking images of grove-bound white-cloaked mystics or woad-painted warriors clashing against legionnaires) and the academic usage, which refers simply to speakers of a particular language family. The popular usage isn't wrong necessarily, it is often inspired by some authentic descriptions, but the two types of usage don't always overlap. In an academic sense then you could say that the inhabitants of what was then Gallia stopped being Celtic when they stopped speaking Celtic languages...which isn't really what you are getting at, I realize.

You have hit upon another really interesting question though! Since ethnicity is wrapped up in self-identity, a great question is how did people in France regard their ethnic history? Did they see themselves as the inheritors of the Gauls, or was it the Franks? Or the Romans themselves as in other Romanized parts of Europe?

These aren't questions I can answer with expertise, but I suspect someone who specializes in French history might be able to do so.