I used a private GP. We first went to him when our GPs refused to refer us to a neurologist for ds2's epilepsy. The consultant ds2 was seeing at the local hospital was only a paediatrician with an 'interest' in epilepsy.
I know we could have fought it and demanded a second opinion but as far as I am aware (and I did look into it) within the NHS, if you are already under the care of one consultant you can ask for a second opinion but there is no urgency in finding one for you.
NHS choices page on second opinions
My son's seizures were not getting better under the regime the paediatrician was recommending and she wanted to increase the drug he was on (which was giving him behavioural problems) rather than trying a different drug. The paediatrician explicitly told us she wouldn't refer him on to a neurologist as his epilepsy was uncomplicated and (according to her) the neurologist would have only suggested the same regime as she was using in any case. However we were prepared to pay money to be seen by a different doctor sooner rather than later.
One thing I found particularly annoying, although this is beside the point really, was that he only ended up under the care of this consultant because we had taken him into A&E following some bad seizures. My older son had actually been referred directly to a neurologist by the same GPs who refused to refer ds2 to a neurologist for displaying symptoms that only possibly pointed to epilepsy (and turned out not to be). Also, the paediatric neurologist we ended up seeing privately, said that in her NHS trust, all children with epilepsy were referred to her rather than to a general paediatrician. It really seemed to be completely unfair that ds2 had ended up with a generalist instead of a specialist.
As other people have said (those who have used private GPs) none of the notes go to your NHS GP at all unless you want them to. I tried to keep them all in the loop and copied all the letters etc. to both places but I found the NHS doctors were all very sniffy about us having gone private and went on about how it wasn't ethical to have more than one GP and we should think about what was in the best interest of our son (as if that wasn't what we were doing in the first place).
Having said all this, the rest of the family stayed with the NHS GP - and why not?
Interestingly, we have since moved to Australia where there is no obligation to stay with one GP's surgery and you can be registered with as many GPs as you like. It makes me wonder about doctors making diagnoses with an incomplete knowledge of a patient's medical history, and indeed the possibility of a patient doing exactly what sollylololo mentioned. But at least you are not stuck with consultants you don't want - and if a GP treats you badly it is very easy to vote with your feet and just never go back there again.