Unless you can approach the U.S. security/intelligence community with some unbelievably rare skill as a non-citizen (like you're fluent in a critical foreign language--Tamil, Pashto, Urdu, Dari, etc. or your background is in the hard sciences a.k.a you're a nuclear engineer or something along those lines), then you're not likely to get much interest from the U.S.
But, if you eventually do go to work for the British government, whether you get to work with U.S. government employees really depends on the particular agency you work for. Read up on the different types of intelligence specializations out there (HUMNT, SIGINT, MASINT, OSINT, etc.) and figure out whether you'd be interested in doing something other than HUMINT (which is what CIA does). If so, I think your chances to work in the U.S./for the U.S. would be better. Read up on the Cambridge Five and you'll see that there's a history of tension between the U.S. and U.K. HUMINT communities regarding espionage. But other U.K. agencies may have better working relationships with the U.S.
Bottom line: If you want to do it, it would take several years and you'd have to acquire the right skills and experience. There are intelligence community employees who come from all over. They've renounced dual citizenship or become naturalized citizens in order to be U.S. citizens and get their intel jobs. But you can't just walk in the door.