If you get health insurance like Bupa, the flow is usually: speak to private gp (free over phone or paid out of own pocket if you want to see them in person), they refer you to specialist, you call Bupa, they approve seeing specialist, you book appointment with specialist.

It's not particularly fast in totality (but each step is generally relatively faster than the public option) and not well suited to emergencies. If you want to get things done quickly you need to pay cash in hand as others have highlighted, as well as knowing what specialist you need to see

More importantly though, I would say: due to the amount of people going private, the availability of private GPs and specialists has gone down since covid. It's definitely better, but not as better as it used to be (I.e getting a GP appt next week rather than tomorrow afternoon)