There is a bit of a myth going around that US scholarships are sooooooo generous. Maybe. They are very difficult to get. Student loans in the US are complicated. It costs more to go to school so far away from home as well. US schools are 4 years rather than 3, and your parents are very, very far away. I am not saying don't do it. But some of the stuff I read in the papers is just ridiculous.
I think there was a sweet window in time about 10 years ago when kids trying to get into Oxbridge who didn't make it were easily waltzing into Ivy League schools, and their parents were delighted to find that there was some financial aid on top of it all. I think those days are gone. I think the schools didn't have many British applicants and were delighted to have a few token Brits for "diversity." The schools don't need to try so hard to attract Brits anymore. Also, these were generally well-healed families who were still shouldering high costs even after partial financial awards.
I have a friend in the know at Berkley in California. He tells me that tuition is now well over usd 20K per year for in state students. And that the cost for foreign student is even higher. Foreign students are a delight because they are cash cows and there is ever greater pressure to award them places over California residents because they are so lucrative (sound familiar anyone?). There is much sole searching about the mission of public universities in the USA. Is it to educate the citizens of the states they are in, this being their original mission? Or to simply pursue academic and intellectual excellence without regard to the society outside their campus? The tenured profs with a professional stake are plumping for the latter, and there is a full tilt towards going for the money.
Also, there are genuine social issues with gender politics on undergraduate campuses in the US. I would definitely keep my daughter away from large universities with Greek systems. (I expect a flaming for this, but mum to mum, I really, really, really would.)
I would be much more comfortable sending her to a small liberal arts college, but that doesn't sound like it fits in with her aspirations.
So, all that said, earlier posters who suggested waiting for graduate school, going over as part of an exchange from a UK university, and looking at Fulbright Scholarships are spot on in my opinion.