The NSF does follow you overseas - the only difference in that case is the tuition allotment is limited to $10,000/yr. In the US, schools generally cover tuition expenses above this themselves.
Another good program to look into is the Marshall Scholarship. It's specifically in place for US students wanting to study in the UK (and vice-versa). Scholarships generally last for two years, though third-year extensions are possible.
Have you looked into funding from the Research Councils? In general, they only fund students from the UK or the EU, but if you have close relatives from Europe, you may qualify. Additionally, there is a council for overseas students (i.e. non-European), to study in the UK - http://www.orsas.ac.uk/
. The method for applying for an Overseas Research Award varies from school to school, so you should look into how your school takes applications. At Cambridge they've stopped accepting applications for these awards; I don't know if it's just a university policy or if the program has been put on hold because of recession issues, but it's probably still worth some research.
And finally, I'd suggest once you've decided on a school and program, talking to your department/research group/supervisor directly. Once I'd committed to Cambridge, I told my research group that I was definitely showing up, whether I had funding from the University or taking it all on a loan. When they heard that, they offered to cover ~1/3 of my tuition and fees for the duration of my PhD. It's very unusual to be fully funded from just one place, so do whatever you can to piece it all together.