Most European countries have some kind of hybrid system, where treatment (at least out-patient and GP visits) is not free at the point of use, or at least where the patient does actually see the bill even if they don't pay it. There is then some arrangement whereby one or more insurance schemes either reimburse you, or pay the bill directly. Sometimes this comes with modest co-payments. Spain is a bit of an exception — its system is very close to the NHS.
To take France as an example: A visit to the GP costs (in most places) €30, a price which is regulated by the state system. The system reimburses 70% of that, minus €2 which is considered to be your "skin in the game", so €19. Most people have top-up insurance via their employer that will reimburse the other €9. (It's actually illegal for an insurer to pay that last €2 back to you — the idea is that people with time on their hands need to be discouraged from visiting 10 GPs until they find one who can be bullied into giving them antibiotics for a cold.) But if you have a very low income and/or chronic conditions then you can get 100% reimbursement. And for major things like hospitalisation, where 30% would still be quite a lot, the system covers pretty much everything. Your top-up insurance might pay for a better room but that's about it.
In some countries there is a mixture of private and public hospitals, in others it's all private but the government regulates prices. Even in Switzerland, which likes its capitalism red in tooth and claw, medicine is private but the government guarantees that everybody, regardless of pre-existing conditions and disabilities, can get insurance at an affordable price.
So if you want examples of systems that I either have experience of or know of people who live there and are happy: France, Germany, Switzerland, Netherlands. Of course, all of those systems have issues, but is none of those countries (or Spain) is health as prominent a political topic as it is in the UK.
Now, how you would go about getting to that situation starting with the NHS model is an interesting (and hard) question. Overnight you would have to put in place billing systems and insurance rules, which would add many billions in overhead themselves, and hope that that would somehow allow people and the state, between them, to spend more (% of GDP) on healthcare, which is the real problem of the NHS. (It reminds me a bit of Universal Basic Income, which may or may not be a good idea, but if it was ever introduced it would be incredibly hard to remove it.)