2D says:
You can obtain a full-blown independent FAA PPL. This entails sitting the single FAA Ground exam (a computer-based test that can be sat in the UK), and then going to the US and flying the equivalent of a skills test.
In addition, for the unrestricted FAA private certificate, don't forget that you will need to show 3 Hrs instruction by an FAA instructor in the prior 60 days in your logbook, be signed off by an instructor as suitable to sit the written and practical tests and sit the oral (which some people regard as the more challenging test) immediately prior to the actual flight test.
The PTS (practical test standard) booklets are a very good grounding in what's expected. Personally, if your'e serious about flying in the US, I think it's a great thing to do, rather than relying on the reciprocal licence.