There are quite a few posts in here about private insurance. Typically, though, even with private insurance, you still have to go to an NHS GP for a referral. There are private GPs but not all insurers cover them. So if it's just your local surgery/GP you're having issues with, private insurance is unlikely to help.
If you get a referral then private insurance can be helpful in queue jumping (many NHS specialists also have private practices). Private insurance can be particularly useful with things like physical therapy, chiro, osteo, etc. as there are tons of private clinics for that sort of thing. You can get cancer care through private insurance but you'll still probably end up getting it in an NHS hospital. Private insurers look very carefully at chronic conditions so if you injure your knee, then keep hurting it, they'll eventually stop covering you.
Also, as mentioned you need to be thorough about prior illnesses and conditions. Had a sinus infection? Tell them. Iffy pap? Tell them.
HTH.
