What are your expectations of private care?
Most will require you to be referred by your GP, so if it doesn't cover GP appointments, only secondary care, you'll have to wait for a GP appointment like everyone else.
Most policies follow similar clinical guidance than the nhs, so for instance, my SIL had to get drug treatment, physio before she could get surgery on her shoulder.
Most will also the same guidelines in terms of cancer treatment so you are unlikely to be approve for expensive treatment not available on the nhs. Although there might some few exceptions.
You often have to transfer back to the nhs as the private hospital doesn't the expertise or equipment to undertake the surgery or again, you might not be covered for that procedure. You used to be able to 'jump the queue', it is now much less likely.
Many will not fund what they deem 'cosmetic surgery' ie varicose veins, so look closely at what it covers.
You also have to consider emergency complications as most do not have ICU, so require to be transferred to the closest NHS hospital, adding risks during transfer.
Private treatment is good if what you are looking for is quicker diagnostic imaging, and outpatient appointment, and possibly non urgent surgery.
I personally think you gain more by understanding the NHS, for instance knowing that you can ask to go to any NHS hospital in England, regularly asking where you are on the pathway, and able to speak to the secretary and establishing a good relationship for your GP learning about whatever condition you may suffer from so you can ensure they are following the clinical recommendations and referral process as they should.