Piper.Classique - Your French issued EASA Licence is valid for flying French registered aircraft. You cannot use a UK medical:
https://www.easa.europa.eu/faq/127224.
You'll need to get an EASA Class 2. This can be issued by any AME holding an EASA authorisation. I hold an Irish licence and do my medicals with a Transport Malta AME in the UK.