it depends.
British citizens who move permanently to a country without a bi-lateral health agreement with the UK (the US falls under this category) are actually not eligible for the NHS when they come back to visit. BUT...as soon as they move back to the UK to resume permanent residence they
are entitled to NHS treatment once again.
However, there is a regulation that says British citizens (and others) who have lived in the UK lawfully for 10 continuous years and then move abroad to work overseas are still eligible for the NHS for five years after they move away. So basically, most Brits are still eligible for the NHS for the first five years of living abroad and then after that they lose NHS privileges, unless they live in a country with a bi-lateral health agreement with the UK (in which case they retain partial privileges.)
In practice, though, I doubt most surgeries are going to bother to find out exactly what a British person's permanent residence is...if it's going to involve a lengthy hospital stay/expensive major surgery it might become an issue, but if a expat Brit on holiday in the UK gets the flu and needs to see someone I don't think anyone's going to ask them to pay.
Full guidance
here.
