Can't log the time though.
You don't need an M1 anyway.
You need to apply for an FAA PPL based on your foreign (JAA) license.
Steps:
1) Fill an authorisation form for the CAA to release your details to the FAA and pay £38
2) Apply for FAA license based on your JAA license
3) Wait for response
4) Book interview/appointment with the FSDO that you choose. This will be in the states
5) Go to interview, fill in more paperwork, and walk out with a piece of A5 paper saying temporary airmans certificate. Free and valid for 120 days.
6) When you plan to fly in the states go to any flight school and do a BFT
7) Bobs your uncle
8) One day your FAA green credit card style licesne will arrive at your home.
9) Every 2 years do a BFT