The short answer is, yes. It’s fine to see a private GP if you want to and, if necessary, they will refer you to an NHS hospital consultant (or to see that consultant privately, if they do private work).
A private fully registered GP (general practitioner) can see you and provide ‘pre-referral’ services to assess your needs. Then, if necessary, they can refer you to the NHS for further diagnosis, tests, treatment or surgery.
As explained by the British Medical Association (BMA; one the doctors’ unions), guidance from the Department of Health and Social Care (DHSC) says:
‘Patients who have chosen to pay privately for an element of their care are entitled to receive NHS diagnostic tests free of charge as long as they are eligible. A referral by a private GP for an NHS diagnostic test should not be any different from an NHS GP referral.’
Patients do not need permission from an NHS GP to go to a standalone private GP service.