Write to Tesla in the UK and see what they say. If they decline, I’d probably try a small claim in the UK- you don’t have much to lose. Just do your homework on who your taking to court (and why). Tesla will probably either settle in advance or just no show.
Legal jurisdiction hard to say without your repair quote/invoice/car purchase paperwork.
There is a Tesla Spain (Tesla Spain, S.L.U., Calle de la Metalurgia 38-42, 08038, Barcelona) assuming this is a separate legal entity.
You probably bought your car from Tesla Motors Ltd or a finance company did in the UK.
The warranty is with Tesla Motors Netherlands B.V.
Probably all wholly owned subsidiaries of Tesla Inc.
One line of thought: You booked the appointment with your UK app - did Tesla effectively contract the repair to a different subsidiary. Think principal contractor. They could have said ‘no’, you’ll need to get the car towed back to the Uk under breakdown.