This page from Frontline gives a brief discussions of how five other countries — UK, Japan, Germany, Taiwan and Switzerland — cover their populations and links to much more detailed discussions of how each country does it with experts on how that particular country’s system works.
As others have noted, the Switzerland model seems closest to what health care reformers in the US are now trying to create. Another interesting point is that the Swiss system was only put into effect in 1994, which is significant for a few reasons.
There are cultural and political-economic roots behind the fact that the US is the only advanced industrial country without a national health care system of some sort which guarantees health care coverage for all. But one point that may get too little attention is that almost all the other countries put their systems into effect during the heyday of social democratic reforms in the first half of the 20th century. Fairly few countries have done this recently. It’s a very good question, with a lot of complex answers, as to why the US didn’t do it then. But what probably gets too little attention is that medical care itself is radically different than it was then. And as an industry — with vastly powerful insurance and pharmaceutical industries — it all but didn’t exist.
What this means is that the kinds of moneyed interests which now oppose reform simply did not exist in anything like the same form. And that is a very important difference. Yet Switzerland managed to pull off significant reform in an era with a big medico-industrial complex.