I currently live in Dallas and want to move to London. I am going to try to do this on a work permit but I don't know how easy that is. How many London employers hire Americans who want to live there?
It's very difficult to get a job that would get you a work permit and bring you here. My husband has a Ph.D. and teaches in a field where there is only a limited number of people who have teaching experience in that area and are at his particular level on the university ladder; that's how we ended up in the UK. You need to fill a need that can't otherwise be filled in England or the EU, and that usually means you are highly skilled in some way -- a doctor, nurse, something in the computer field. You can also come on the points system as a highly skilled migrant -- and you will need a high-level degree and several years in your field, preferably in a management position. (I have a bachelor's degree and 12 years in a professional field, and I wouldn't qualify as a highly skilled migrant on my own).
One other option is to find a U.S. company with offices overseas that would hire you and send you to the UK. This also isn't that easy to find, and probably would be a job that required a lot of experience -- and the people already working for the company would probably get the shot at a coveted overseas job before a new hire would.
And the NHS isn't all it's cracked up to be by people looking in from the outside. It's actually no better than the U.S. system, it just has different strengths and weaknesses than the U.S. system.
All that said, yes, you'd be covered by the NHS if you were here on a work permit. You are also covered if you are a student.