I live in Quebec and technically I don't have nationalized healthcare but rather health insurance. It doesn't change the fact that I don have to pay anything out of pocket at any hospital I go to here.
An itemized and detailed claim will be made on my behalf to the insurance agency to get everything paid.
From a user pov it makes no difference how the system works.
Quite the contrary, the level of service can vary wildly in nationalized health care, where the state runs the hospitals. There are pluses and minuses -- you can get much better health care at a much cheaper price to the tax payer, but with all the problems you see in any government bureaucracy.
Nationalized insurance on the other hand is pretty much the system any American would be used to, except the government pays and you have long queues.