"the most serious medical needs get treatment first"
If that's the definition of universal health care, no country I know has it, because there is no country I know of that has successfully banned cosmetic surgery.
I think that might be a usage thing: over here in the UK they're both plastic surgery, but you see it qualified as reconstructive plastic surgery or aesthetic plastic surgery.
Still, take birthmark removal: it's purely cosmetic, in a sense, but it makes such a difference to peoples' quality of life psychologically that it totally should be on the NHS if you ask me.
If that's the definition of universal health care, no country I know has it, because there is no country I know of that has successfully banned cosmetic surgery.