I don't believe that govt. paid universal health care will work in the US the way it works in Europe. Those of us with power/money (i.e. good health insurance and a modicum of education) expect excellent treatment on demand. I am pretty sure that Americans--middle class (shrinking though it is) and up are not willing to give up that access and most of us believe that we would be giving up some access.
Having been treat both in a country with UHC, and a country without....
In the US I was called sir, the waiting room was extremely pleasant, I was treated almost immediately after they'd checked my insurance details,, and the customer care was excellent. The latest drugs were available and were prescribed in large quantities. They had some concerns, and I was offered a consultation (i.e. with a specialist for my condition) for three weeks time. That wasn't soon enough to actually do anything about my condition (blood poisoning), but that was the best my insurance could manage.
In the UK I was in a drab waiting room, just an alcove off the main corridor. There was no TV, no refreshments, and the person keeping track of us was also tracking the doctors and nurses, which unashamedly took priority over us. There wasn't really any 'customer care' as such. She was treated after a wait of just over an hour, and was referred to a specialist the same day. This required yet more waiting (3 more hours, since they weren't on site until the afternoon), and then she was seen, diagnosed, and given an appointment for the following day.
So the US wins out on drug availability, customer care, speed, and general quality of experience. But in the UK I got the medical treatment I needed, at the time I needed it. My wife benefited from the speedy treatment by a specialist, whereas the prognosis would have been much worse if she's had to wait on specialist diagnosis. If I had been in the US, she would have needed surgery. In the US and without insurance, she would almost certainly have died.
My impression is that access to care, particularly for dangerous and life-threatening conditions, is far better in the UK, even for those with a fair amount of insurance. Certainly the treatment of obscure and difficult conditions is better. But you won't get a TV, you won't get much customer service, you'll have to wait a number of hours before and during treatment, and during that time no one will try and sell you anything.
I know which I'd rather have.