If you are a legal resident, you are covered. If you are a visitor it depends on what deal your national UHC system has with the nation you are visitng. Normally there would be little reason to go to a foreign country for treatment.
Most UHC countries have reciprocal deals where a citizen of one has all the same rights as a citizen of the other when travelling. Its just a lot cheaper than having to ship people around or delay treatment or all the paperwork.
The US has no such deals because it does not have a UHC system to deal with. However, most UHC systems that do not charge aren't going to pay out for setting up a billing infrastructure for the occasional American. The ones with co-pays etc already have a billing structure in place, and may charge you. Or not, it would be pretty mortifying to have to charge someone for health care.
The NHS is what is called a Beveridge style system. I think that is what you mean by single managed. Similar systems are used in Spain, the Nordics, etc. They are often managed at local levels though and may include private hospitals and treatment that gets reimbursed by the payer at the same level as public ones. The VA is an example of this type of system although other nations have far more oversight and performance control than the VA does.
A competing setup is the Bismarck style of insurance company administered management, either for-profit or non-profit. Germany, the Nederland's, Switzerland, etc runs this sort of system. Always with a lot of legislation and safeguards to ensure that people do not fall outside the system. It is similar to the US employer-provided insurance setups.
Other systems are National Insurance, as used in Canada and Japan. Similar to Medicare. Canada is fairly unique in its hostility to private provision.
And then there is out of pocket, like rural Africa does it. That's basically like being uninsured and not qualifying for Medicaid.
Thing is, the US accumulated massive bureaucracy and costs by h\trying to run every system at the same time with little attempt at standardization and lots of billing and gatekeeping. Everyone else picked one system and stuck with it, letting private providers fill in the gaps. (Except Canada for the last)
The VA is much the same as all the other US healthcare setups. A system with the controls taken off, trying to operate in a jungle of different systems and bureaucracies.
IN most nations with a VA-style (Beveridge) setup, there is great media interest in the least failing or hint of scandal. And they mean politicians lose a lot of votes. The VA hasn't enjoyed that feedback system until recently.
