I don't know any pilot that being non-uk born is working for Flybe or BA for example ,
...Can we put the BA stuff to bed?
I can't comment on current recruitment but last weekend, having said hi to a Dutch national BA captain who has been in BA well over twenty years I then.spent an hour talking about the Shoreham accident with another BA captain who is a US national (been in BA 20 years'ish, before you ask as right to work live in EU /UK due family circumstances.)
I've just done a three crew long haul trip where neither of my colleagues was a Brit, the checker on my last sim wasn't a Brit. It's certainly not unusual to have a least one none Brit on the Flight deck on a trip..and I haven't yet mentioned the Irish, probably because there are too many to count....