This is a terrible idea. Also, the NHS is not free - we pay for it through our taxes. It's simply free at the point of use. There's an inherent unfairness for super healthy, non-clumsy folks, but, when you need it, it's there - and you would want to be able to have that support and treatment, rather than worry you can't afford for your broken ankle to be sorted, for example.
So, I had to look at private healthcare recently, as there was a massively long waiting list for some significant surgery I needed (total knee replacement.) To go private - because, of course, existing conditions aren't covered on insurance - it was almost £20k. I was looking at going to Lithuania, of all places, as it was about 1/2 that price, if I had to go private. (Also, incredibly high standards of care, training, etc., there; I'd recommend it over Turkey, as also a shorter flight....) Anyway, in the end my case because urgent, and I got the surgery here, thankfully.
I would have been happy to pay for a single room, nicer food, etc., but the key thing was the quality of care. It's having the follow up, any questions you might have, and all that; all of which is also chargeable privately.
I did have private surgery a few years ago (covered via the NHS waiting list initiative), and the care was no better - they were just better staffed, there were plush private rooms, and the food was much nicer. The surgeons, nurses, physios, etc., provided just the same standard of care.
Anyway, I would strongly oppose paying for the NHS, as it would stop so many people from seeking the help they need. Think of the times where you've had to keep going back to get something sorted - or for vulnerable, chronically ill, disabled, or older people - and it would be a barrier to doing that. I don't disagree with a system for charging for non-attendance; BUT you'd have to make sure that the administration of it all wouldn't outweigh the income. That was a big driver in doing away with prescription charges in Scotland anyway, because the system generated so little in comparison to the process of exemptions. (Can't find the research/evaluation report right now, but it was a Scottish Government evaluation one of my besties worked on; she's not a politician, but an academic, before anyone points that finger!)