It's not done by a flight school, it's done by the FAA. You provide them with information, they contact your CAA to verify your license and you go to a FAA office to pick up your certificate. Since you're a Brit, your CAA wants paperwork saying the FAA will be asking about you. After you get your certificate you have to complete a flight review with an instructor to be current. The regs require a minimum of 1 hour ground and 1 hour flight time for the review but I'd plan on several hours ground time to get up to speed on the differences.
Airmen Certification ? Verify the Authenticity of a Foreign License, Rating, or Medical Certification