You're going to be a resident of France, and as such you will have one month to re-register a car with foreign plates which you take with you.
Your UK driving licence is valid in France until it expires, while you are allowed to keep UK plates for up to six months if you are a non-resident of France, and up to one month for French residents. The time allowed for French residents can be extended up to a maximum of three months should a vehicle need to be approved via the DRIRE*. In the past these time limits have not seemed to have been strictly adhered to, but insurance laws are now tightening and this should no longer be the case.
* Direction Régionale de l'Industrie, de la Recherche et de l'Environnement
I think this official site is talking about the same one month limit (emphasis in original):
Avant de faire circuler pour la 1ere fois votre véhicule d'occasion acquis à l'étranger (importé), vous devez dans le délai d’un mois, sous peine d’amende, faire établir le certificat d’immatriculation, aussi appelé « carte grise ».
Disclaimer: note that I haven't lived in France and I don't know whether you will have some margin due to not being officially resident until some other bureaucratic process has finished. Hopefully someone with experience in the matter can give a better answer, but I was alarmed by some attempts to answer in the comments which I know to be wrong in Spain and which I fear could potentially open you up in the worst case to criminal charges by causing you to drive without valid insurance.