Funny thing: in the last two countries I've lived in, you have both a full National Health Service as well as private medicine if you can afford it.
They appear to work well, and for many years here in Spain I paid into a private scheme for my wife and myself, though we were eligible for the state-funded one too.
She broke her hip one day, and was rushed to the local state hospital because it was half the distance away than was the private one we used for her cancer.
The service and quality of care was so good, that she suggested we should get over it and stop the private insurance. That was ten years ago. My last premium was €3600 per annum, so I guess that the state has saved me €36,000 on the clear understanding that inflation in medicine does not happen! ;-) On top of that, the doctors interchange quite a lot, and you can find the same guy working in both, as we did.
Fortunately, so far, I can't say anything based on experience about the UK version, other than that my granddaughter now wears her stethoscope within it.
The costs of public health are borne by taxation; you pay nothing at Reception before or after they save your life.
Strikes me that choice is a damned fine thing to have: if you have money to burn, go ahead and light the Zippo and free up beds for the not-so-well-off.
The funding, as I say, comes from taxation: why is the UK so keen to phase out the petrolheads' delight, and usher in electric cars? Since taxation is the massive part of gas pricing in Britain, does the coming reduction in petrol sales tax revenue indicate that electricity will have to go up in price to match the lost revenue from oil? Interesting ... you read it first in LuLa.