Can I get healthcare in both countries? Can I return to the states to see my old doctor or am I covered only in the UK?
Access to the healthcare system often depends on residency rather than citizenship. Thus all UK residents have access to the NHS (and pay for it through their taxes). Similarly, in many countries on the continent, residents have to insure themselves through the national health insurance system or show they have some other type of coverage to become residents.
There are also some reciprocal agreements between countries that make it possible to get care more easily in foreign countries but that only works if you are already covered by one of the countries in question. For example, if you are a UK resident and are covered by the NHS, you can get a European Health Insurance Card and use it elsewhere in the European Union. But if you are a UK citizen residing in another country, you cannot do that (there are however a few similar agreements between the UK and non-EU countries).
Furthermore, outside of very specific cases (like cross-border workers), it is typically not allowed to travel especially for a medical procedure or to use the system to get routine care outside of your country of residence. Much like private travel insurance, it's mainly intended to cover you if you become ill during a visit.
Being a citizen is helpful to the extent that coming back to reside in your country of citizenship is always possible (whereas foreign citizens might need to show they do have health insurance and/or income to become residents). A few months after you become a resident, you will typically have access to the national health care or health insurance system, possibly free of charge if you are destitute. But you can't just hop in for a medical procedure.
The UK NHS (which you surely have registered with already if you live in the UK or you will when you settle in the UK) does not cover visiting the USA irregardless whether you are a US citizen or not. Now, the US insurance is a very complex matter in general you will not have healthcare in the US unless you live there. If you are employed by a large USA company and are temporarily sent to the UK (for a few months etc) it is possible you are still insured in the USA. Ask your employer if this situation applies.
But typically US citizens living abroad and visiting the USA need travel insurance (example).