The NHS entitlement is complicated, but essentially unless you're here on a visa (not a visitor's visa though) Americans are not entitled to free care from the NHS and would be expected to pay. There has to be a reciprocal agreement between the UK and the other country, unless it's an EU country. If there is, visitors can have certain, necessary treatment for anything that becomes necessary during a visit.
At the moment there is no agreement with the US.
I'd suggest ringing BUPA and asking if they offer emergency services....