I can give you my take from my personal experience here in Belgrade. Short story: public care is bad, private—surprisingly good.

I live in the US and my experience with the Serbian health care so far has been good with the private and not so good with the public care. I use private for my wife and public for my mother. Big difference in every respect.

Yes for the public health care, NO for the private. We walked into a private hospital and got the treatment within 30 minutes, although they claim that appointments are needed.
Private labs are available (some are open 24 hours a day) we got the blood results within 3 hours (emailed to us). Paperwork is almost nonexistent as compare with the US.

The cost are considerably lower than in the US. There are reimbursements would depend on your insurance I was told, I payed for visits my self so I can not tell you about the reimbursements from the personal experience. Pharmacies are all over the place, we were able to purchase drugs without prescriptions, we just showed the an empty box of the medication and they did not make a fuss.

There was really no delay to speak of in the private hospital, I can't tell you if this is the case for all procedures (huge difference there as compared with the US). Lot's of delays and BS in the public hospitals but I guess in line with other countries public health care.

I think I answered some of this above. I am not familiar with the private health care insurance in Serbia, I can look into it if you need me to but I think you would be better of paying cash or with your foreign health insurance, just make sure they will take it here before you commit.

Hope this help and good luck!