Healthcare is all free to the patient (the one caveat being a small, fixed charge for prescription medication - which is free for some groups), all paid for via national taxes based on wealth. UK.
If we need a specialist Doctor, we are referred to one. There's no money involved for the patient whatsoever.
Attaching an unaffordable fee to healthcare would be a clear barrier for anyone who is not upper class, and this would be seen as deeply discriminatory and thus unacceptable.
There is also a private health care sector, with its own hospitals. A lot of consultants work in both the public (NHS) and private sectors, e.g., one day a week they will have a private clinic at a private hospital. The private sector is funded by insurance, and this is often a perk offered by employers. Waiting lists are generally shorter in the private sector, but, in my experience, the expertise and level of care is no better than the NHS.