There are probably a bunch of different options here from no national involvement in health care, to nationalized insurance, to full out nationalization of heath care. In any case its a very tough nut to crack. On the fully nationalized side you potentially run into what Andy D mentioned where everyones on a wait list (in the event of the system being under-funded), or conversely you end up spending way more than you need to (if you overfund it). On the fully private side you have people who are effectively on wait lists until they get insurance, which is never in too many of cases.
The fundamental problem as I see it isn't really nationalization or not. Its more how do you get people to make cost conscious health decisions when they aren't spending their own money. Whether your insurer is paying, or the government is paying, you as a patient have no incentive to reduce your spending. Further, people obviously want the best chance at living no matter the cost, so "the more tests and pills the better" is the motto for lots of people. How can you have an effectively running health care system with that attitude about spending?
If you could gear a system that rewarded you (with tax savings or lower premiums or something) based on how "frugal" you were with your health care you could potentially instill the right attitude in people. That way they could at least make their health care decisions in the context of a perceived cost, and start driving the prices downward.
Until people take personal responsibility for the cost of their health care, either system (nationalized, or private) is doomed to be too expensive, and out of reach for some set of people some or all of the time.