GP’s are all private companies that have very lucrative contracts to supply services to the NHS. They are not NHS employees. THey hire and fire their own staff and can choose how many appointments they have, when they are etc etc.
Im sure you can get BUPA for £20 per month. However it will only cover a limited number of procedures and only upto a certain cost. No private provider in the UK provides emergency care, Intensive care or A&E services as it is very expensive and not profitable. At the moment there is still a high chance that you would be treated in an NHS hospital as anything more than very simple stuff the private hospitals don’t have the support services to cope.