Would this be possible?
Yes.
Do any regulations exist about the training having to be done in the same country where the licence is going to be issued?
Only that the issuing National Authority must hold your Medical records, so you would have to have a UK CAA issued Medical.
Would I be required to sit any courses?
You would have to satisfy a UK ATO/RTF (Flying School) that you have completed sufficient training to be put forward for the Written Exams, the Practical RT Test, and the final Skill Test.(ST)
Do I have to go through a UK flight school?
Yes.
Would CAA be OK with it
Yes.
Make sure that you have your Logbook entries certified correct by the French ATO/RTF, and bring a certified copy of your Training Record and Qualifying Cross Country certificate with you. These may have to be translated. Your UK ATO/RTF will need these to check that you have completed the required flight training, before signing your Course Completion Certificate.(CCC)
You must have a CCC
before the ST.
MJ