As a UK ex-pat in the US I did a bit of reading up on this.
The US/UK have a tax treaty, in fact on your US tax forms it asks for "Foreign taxes paid", so they are not charged again in the US. I'm pretty sure this is a non issue but you will have to figure out how to set it up.
Reading on the NHS website I saw that it said you simply had to prove intent of living in the UK to have full (and immediate) access to the HC care system.. Its specifically stated some examples or proof such as shipping documents for household stuff, rental agreements etc.
I would be a little more concerned about what curveballs Uncle Sam would throw at you to be honest, not that I am aware of any. As a UK citizen you will be required to get a "Sailing permit".. The very notion of that fills me with horror, but I think its just a case of proving you don't owe the Government anything.
A friend of mine (also a UK expat) went and got his US Citizenship as his meant he could avoid having to get a sailing permit should he ever decide to go back. It also means you don't have to keep renewing your green card every 10 years. As mine expires in 2019, I might do this as well.
I went back to the UK this Summer and to be honest I was shocked at how expensive things had become, for some reason it didn't feel like home anymore but I guess after almost 20 years in the US that's understandable.