i think it varies. for UCL, you can apply if you are doing a masters or a masters with a phd following. for LSE, it's only for one year masters students, so i guess it depends on the university. i saw these requirements on the uk fulbright website.
http://www.fulbright.org.uk/fulbright-awards/exchanges-to-the-uk/postgraduates
the partnerships have so many less applicants, making your odds much better. last year, there were 4/271 chosen for the open award, and UCL is 1/44, or Imperial is even 1/20. some on there i saw were even 1/4.
http://us.fulbrightonline.org/countries/selectedcountry/united-kingdom?program_id=218