With a public system right there, it makes no sense for private providers to waste a penny on anything that doesn't generate profit. So e.g. if you have an emergency while you're physically at a private hospital, they will almost certainly phone for an NHS ambulance, because the NHS does emergencies and they don't. Heart attack? Not their problem. Broken arm? Not them. Car accident? Nope. Those can all go to the NHS. But if you need a routine or even cosmetic minor operation, they're happy to take your money and offer you a slightly nicer experience, air-conditioned room with fresh flowers, branded products instead of whatever was cheapest for the NHS to source, that sort of thing.

Some of the private schemes are literally insurance. So you get the exact same NHS treatments as everybody else, but your insurance means e.g. you get a $50 cheque for the inconvenience of seeing a doctor to treat that nasty infection, that sort of thing. This is a bit trickier for them to implement since obviously the NHS doesn't want its doctors doing paperwork for some insurance company on tax payer time, and the insurer isn't keen to pay a doctor's hourly rate to fill out a form saying "Yup, your customer had a nasty infection, I prescribed ointment". But if you really wish you got a cheque every time you spent time in a doctor's waiting room I guess it's an option.

The free one is all you have. The insurance from work covers drugs, massages, eyeglasses and dentistry.