In brief - I live in Serbia long-term so am full-time (self-)employed and pay my own contributions, and therefore have full access to state healthcare. There are many shortcomings in the system, but many of them could be applied to British or other national health systems too, albeit in a more extreme extent: a lot of waiting sometimes, difficulty getting specialist appointments within a reasonable time-frame, sometimes rude or just unhelpful doctors and nurses who don't have the time or inclination to give you a detailed explanation of their diagnosis etc. But it could be a LOT worse - it IS basically a modern health system, and works pretty well in the big cities.
In particular I had one family member develop cancer, and he received all the necessary chemotherapy, according to the latest protocols, using treatments that I know cost many thousands of euros, all on the national health service.
I myself had an accident where I broke my hip/femur, and was received promptly in the (VERY modern) Novi Sad emergency centre, and was operated on in a space-age-looking operating theatre within just a few hours, and the surgeon (Iranian, as it happens
) did a very good job, I might add. I was given follow-up checkups (though waiting times for those were VERY long, hours sometimes), and was also called in to spend 2-3 weeks in the rehab ward, doing physio exercises to get back into shape. I was also offered "banja" treatment (health/rehab resort) after that, which would also have been covered by my insurance (I turned this down, too much time off work!)
So the national health service does provide a lot - but inevitably you will find yourself also going to specialists privately and paying for certain checkups, just because they are either unavailable on the state health service, or waiting times are too long (say, colonoscopy for screening purposes). However, these are not too expensive by western standards.
OK, enough from me - I would just say in brief that Serbia does basically have a modern, Western health system for all its shortcomings, and in my opinion you should not let healthcare be a factor in deciding whether to come here.