Where I live there are very few doctors, nurses, dentists or veterinary surgeons who have English as their first language.
I remember 40 years ago when you could get a same day appointment for your GP or vet and i wasnt aware of any non English ones but now a non urgent appointment for doctor, dentist or vet is 3 weeks or longer and it is never someone with English as their first language.
I don't know if Brits qualify and leave or we just aren't training enough but without immigrants UK health services would collapse