I think the majority of non-third world countries have some form of socialized medicine. I consider medicaid/medicare a type of socialized medicine, though not compulsry.
The difference between the proposed US health care system and the current systems of Britain and France is:
Great Britain: National Health Service, or NHS. Gov't pays 100% of medical bills from the beginning. Doctors have set amounts they get paid for each service they provide.
France: Gov't reimburses the patient 70% of the price they paid. If you make below 6,600 euros a year, it's free. Doctors get paid immediately by the patient, instead of waiting on the gov't to pay them.
The list would be shorter only to list those who do -- bc think of all the third world countries.