Maybe its hard to actually have a definition for universal access. In a way the US has universal access via emergency rooms which are required to give service regardless of people’s ability to pay.
I guess universal access means that most/all non-elective surgery is paid for by the state via taxes. How many countries on earth have a healthcare system like this where most healthcare procedures are paid for indirectly via taxes?
I think England and Canada have similar health services, that cover most of the population.
Israel has one of the best—a flat tax (5% of salary) provides full heath care to everybody, from flu shots to heart surgery.
Off the top of my head:
The UK, France, Spain, Portugal, Germany, Sweden, Norway, Denmark, Finland, Israel, Canada, Hong Kong, Singapore, Japan, Australia, Cuba (of course!). Probably many more. Aha - Wikipedia article about it.
Ireland has a hybrid system that is both state funded and paid for (subsidised) at the point of delivery except for the very poor.
Strangely, communist China doesn’t have one.