At Stanford, we can go anywhere we want to go. We can go IN-STATE (UC Berkeley, Harvey Mudd College, UCLA, UC San Diego), OUT-OF-STATE (Columbia in New York, Rice in Texas, Harvard in Massachusetts) or INTERNATIONAL/out of the country (Warwick, UK; Oxford or Cambridge, UK; University of the Philippines; National University of Singapore; University of Tokyo; Beijing University; Grand Ecole, France). The choices are infinitesimal, so to speak, so long as the student is willing to extend his program length in case the university/school he intends to go to does have a different school calendar.
I believe Warwick has as much wide options as has Stanford’s; you just need to inform your college so that in return, your college (Warwick) will notify the university where you intend to attend.
However, please bear in mind, that some (or maybe a lot) of US schools, especially the very prestigious ones such as HYPSM plus Wharton, Caltech, Haas-Berkeley, Williams and Amherst, are very competitive for exchange student applicants because aside from the limited slots available for exchange students, the number of applicants are quite huge.
As a head’s up: Stanford is very friendly to exchange students as the school encourages its students to participate in the school’s exchange student program. A number of Stanford students attend schools in other countries before they graduate. But let me warn you too that Stanford, Palo Alto is an upscale surrounding, so I suggest that if you want to attend a program at Stanford, you must have a good budget and your parents must be very willing to support you financially.
If you don't have that much budget, there are very good universities in Southeast Asia that offer excellent academic training but very affordable to many British students. These are the Universiti Malaya in Malaysia, The National University of Singapore, The University of the Philippines in Diliman and Manila and Chulalongkorn University in Thailand. These are the National universities in their respective countries and they are very prestigious universities in Asia. I believe all these four national universities would be extremely happy to offer you a place as an exchange student from Warwick.
Aside from those 4 universities, there are also excellent universities in India and China. I'm sure Warwick has got arrangements with some of the finest schools in those 2 countries.
If you're not willing to learn another language because you're English... your best options are the Philippines, Singapore, Malaysia and India. Those are the only countries in Asia where English is widely spoken. In the Philippines, even a taxi driver can communicate with you in English. The majority of the people in Thailand, China, Japan, South Korea, Taiwan, Hong Kong do not speak English and it could be a pain in the @ss to live their for a semester or year. However, those are very nice countries and they have very good universities there too.