I'm not sure (please correct me if I'm wrong), but I think you still have to go to see an NHS GP for everyday problems even if you have private insurance, as the insurance is usually used to get things like operations or special treatment done more quickly and efficiently than the NHS.
When you get to your UK university, you will be required to register with the university health service where you will go to see the doctor or nurses if you need to. The health service will be on the NHS. You can use your private insurance to cover costs of special treatments or for treatments that have a very long waiting list (although I have a feeling that the NHS doctor may need to refer you for this to happen - not sure though, so I'd check with someone who knows more about it), but if you have, for example, a cold or a chest infection or something, you will probably still have to see a doctor at the university health service.