Hi,
I'm an aerospace engineer and an US citizen. I would suggest studying for aerospace/aeronautical engineering, although as you can tell, I'm probably somewhat bias. I think aerospace engineering majors have better job opportunities than physics majors. You can also always change your degree or interest area; I did my undergraduate in mechanical engineering and then did my master's degree and soon my PhD in aerospace engineering. I also honestly don't think it really matters which school you attend for your undergraduates as long you get good grades and get experience through an internship(s) and/or research projects.
If you really want to work for NASA, your best chance to study in the US and get an US citizenship. Almost all of the job announcements I have seen for NASA, or any aerospace job in the US, has US citizenship as a requirement because you need to be able to obtain a classified clearance. I'm not sure on the specifics on becoming an US citizen, but I think you need to live in the US for 5 years before you can apply for citizenship. So if you complete your undergraduate and master's, or master's and PhD, it should give sufficient time to meet the time requirement. Studying in the US, especially for a graduate degree will also look good on your application when you're applying for US jobs. Otherwise, it is really tough to work in a different country, such as the US. I wanted to work abroad in Europe for a few years before I settle down, but it's really tough to get a foreign job without any ties to the country where you're trying to find a job in. I eventually had to settle for a job in the US before I got accepted for a PhD at University of London. Ironically, my friend from Spain is trying to find a job in the US and she having very hard time of finding one in the US. But, it's relatively easy to go to school in a different country for a degree and get a job afterwards. I was able to get accepted to University of London without any troubles and I could have gotten into several other UK schools if they had funding available. On the reverse side, many of my friends at University of Illinois were foreign and they were able to study in the US without much trouble either. Few of my foreign friends that have graduate already have been to find jobs in the US.
As many other people noted, UK also has many opportunities too. Astrium/EADS have many aerospace jobs and ESA roughly does same thing NASA does. Really, the only area of research and work the US has over the UK is probably man space flight. Whatever you decide to do, get good grades and try to get as much internship and research experience as possible.