I don't for a second pretend to understand this all, but FR will continue to be an EU airline so will keep the rights they have now except for internal UK services. (If Scotland leaves the Union that may change). These could be negotiated, otherwise FR would need to set up a UK subsidiary.
The UK & Ireland have a common travel area which was negotiated prior to the UK joining the EU and this includes the right to work in both countries. Presumably this won't change. However it MAY mean that UK pilots could only be based in Ireland. Or be based in DUB but spend many nights away at EU airports. Of course, if FR can sponsor pilots for Irish citizenship problem solved. How long must you be resident in NI to be eligible for Irish citizenship.
U2 would require a subsidiary in any EU country.
Now it gets complicated. If FR/U2 wish to operate a common fleet could they wet lease aircraft as an when required from the UK & EU? This may be for individual sectors so that a plane & crew flies EDI - LGW for U2(UK) then LGW - AMS for U2(EU). Both would presumably have rights for AMS - EDI.
As other posts have said the danger is if someone wants to make a political point. However the LoCos must have political clout and common sense may prevail.
Just my thoughts.