I saw many negative references to the healthcare system in Hungary, yes indeed things take time (compared to the countries below too long probably), but personally my first experience in Hungary was actually quite positive (of course it helps to pay a bit extra, but I am pretty convinced that without things will also be OK).
A bit different from experiences in eg Luxembourg or London (locally bring your own night dress (needed for the operation !), slippers and toilet paper). Still doctors and staff seem to be very skilled and result for me was good (as far as I can see now .
My conclusion monthly health care premiums (even with additional payments if you feel the need to) are much, much lower then they are in Western Europe or the US and you still get good service (even if infrastructure and bureaucracy are from 30 years back). Service (and results) are to me up to standard.
So far comparing UK, Luxembourg and Hungary there is of course a difference (but in my view in service, no difference in result).
If I were a billionaire I would have my own doctors, but with a limited budget I am very happy (so far) with health care in all of the countries above where i have lived. (Drawback I do not speak Hungarian, but that is my mistake (I am too old or stupid to learn)).