The short version is YES, you can use any EASA license to fly any EASA registered aircraft (within the limitations of the license, obviously).
The devil is in the details though.
Under EASA all your eggs need to be in the same basket. So you have to choose which state (UK or FR) is going to be the state responsible for your license, and all elements of your license need to be administered there. If you are renewing your SEP and/or medical, make sure that the examiner/doctor uses the forms (and possibly: language) that's acceptable to that state. Existing (non-expired) medicals and other bits need to be transferred to that state. Google "EASA SOLI request" for details.