We are fortunate to have BUPA Health Insurance with DH work.
With BUPA you are allowed to use a service called Babylon which is run by GP's. Its absolutely fantastic, and you can get an appointment with a GP or Nurse Practitioner to suit yourself, and usually the same day.
If needed they can refer you on to a hospital consultant, and issue a prescription, which you can then take to a local chemist.
I have recently had a dermatology consultation and had to go along to the nearby private clinic, and was issued with a private prescription for antibiotics and also a cream. The cost wasn't that bad at about £25, when you consider the cost of an NHS item on prescription . This all took place within 10days, fantastic service, when you consider the NHS waiting lists.
I think Sparklefizz is right about the cost to see a private GP online, and yes Doodledog they do contact your own GP to let them know what they have done.
I hope this helps, we only have access to this service for a few more months as my DH is soon to retire, and so he will lose his Private Health Care. I dread the loss of this facility when I hear how everyone else is struggling so much .