None of the top 10 colleges offer straight up, no-strings-attached an guaranteed financial aid/scholarship for international students. When I applied for colleges I did not have my permanent residence yet, so I had to tick the "international student" box. I can give you a few advice on the trials and tribulations of an international applicant:
1. If you apply for financial assistance from the start then you will have a tougher time getting in. This is not exactly a hidden catch, and the admission office will explain to you in detail but the gist is that they won't let you in unless they are 100% sure that you can afford it. And since colleges tend to have a small pool of money reserved for international students and many applicants, they can only accommodate a certain number.
2. You will not qualify for 90% of merit-based scholarships. I went through hundreds of external and internal scholarships, all of which require citizenship/permanent residence.
3. The Ivies actually cannot give out merit-based scholarships due to some stupid agreement they had sometime ago.
4. Stanford does have merit-based scholarships that are open to international students, but feel free to compete with people who've won International Math Olympics and the like.
5. MIT is more stingy compared to the Ivies, I don't know why but it's just the way it seemed when I called their financial aid office and they were much less ethusiastic on helping me to meet my financial needs.
And one amazingly good news, Caltech offers a huge scholarship that you can apply after your first year (I think). It IS merit-based and it IS open to international students, but you gotta get in first and maintain excellent grade. It's not nearly as ridiculously competitive as Stanford and the like, but you probably have to work just as hard to get it.
So, Caltech is, in my opinion, your best bet. If your family makes 35k dollars then you will have trouble, 35k pounds not so much. Keep in mind that you are NOT in the same pool as americans/green-card holders and you will face discrimination because of your status. However if you say that you do not want any financial assistance, you will be placed in the regular pool and have the same chance of being accepted, but you will face (obviously) a huge financial burden.