These days, in the NHS, people rarely get to see the GP they're registered with. I'm registered with a specific doctor at my surgery but I haven't had an appointment with her for over eight years. When I phone for an appointment, I take the first appointment available, no matter who it's with. Some patients insist on seeing the doctor they're registered with even if it means a long wait. Most of us gave up on that practice a long time ago!
I agree that if you're going to A&E (Casualty) because it's an emergency, you would use "they" to refer to the whole department.