I did this in Florida last year. You need to do the following:
1. Get the CAA to tell the FAA that you're bona fide. It costs £40 - see the CAA website for details.
2. Once that's done, the FAA will send you a confirmation letter.
3. When you've got the letter, you need to book an appointment at the regional flight safety district office for where you are in the US; see the FAA website for addresses.
4. Go along to the FSDO apppointment with as much ID and contact numbers as you can; take passport, driving licence, birth certificate - everything! Plus mobile and landline contacts.
5. The FSDO will give you a temporary airman's certificate which is in lieu of a full US PPL. The full licence takes around 120 days to arrive.
This all takes time; plan ahead and do it now!