I think generally you’ll want to talk with your school and/or university for a student exchange program – it’s not, to my knowledge, ever run by third-party companies. At least, not in Australia – I couldn’t really speak for the UK with any certainty. That said, one I can suggest (if you need something to offer to your school/university) is Ritsumeikan in Kyoto. I was wanting to attend the winter program this year, until I realized I just plain can’t afford it right now. You’ve missed the deadline for the winter program, but you can still apply for next summer:
http://www.ritsumei.ac.jp/eng/html/admissions/program_jp/rsjprwjp.html/
The JET program, incidentally, is not a student exchange program.