For the flight training you appear to be no longer restricted to UK-based dual certified ATOs. The UK CAA appear to have dumped the application of both the 'usual place of business' rule and the 'you must do your IR test in UK airspace' rule and are now approving entirely continental-based ATOs. I have this both from the owner and CFI of an EU based ATO seeking UK CAA approval but do not yet have confirmation from the individual at the CAA who appears to have made this decision. Worth noting that EASA themselves dumped the application of the 'usual place of business' rule when approving UK-based ATOs directly at the Brexit point.
I imagine that some UK-based ATOs will be more than a little miffed at the IR decision.