Without an agreement of some sort between the U.K. and the EU 27, EU airlines such as FR will not be able to operate domestic U.K. flights, or flights between the U.K. and the EU27.
If the most recent U.K. - Eire Air Services Agreement is the default, then they likely could operate U.K. - Eire flights.
But both sides seem prepared to leave such decisions to the last minute.
Oh, if the U.K. doesn’t make a new ASA with the US (and 70 odd other countries) then it could too revert to Bermuda II. Anti-trust exemptions would also expire.