Yes, you can validate your UK license to a US license with private pilot privileges based on foreign license. I did it, based on a Swedish issued CPL.
The process is basically:
1. You fill in an FAA form, forgot the number, but the name is "Validation of foreign airman license" and send it to the FAA with a copy of your UK ATPL
2. FAA sends a facsimile to the UK CAA, they validate the truthfulness of your license
3. When your license is validated, the FAA sends you a "letter of authenticity"
4. When you have this letter, you go to the FSDO (regional FAA office), fill out a form called 8710 - "Airman certificate or rating application". Your license is issued.. done
You should be eligible for an FAA ATP if you already have an ATPL issued by a contracting state, providing you meet the hour requirements of FAR 61.159.
What you need to do theory wise is an "ATP aeronautical knowledge test". I've not done the FAA ATP, but the knowledge test for instrument and commercial are a piece of cake. It could a good idea to study the FAR a bit before you try. Don't know if you can do it in UK. Flying wise what you need is an ATP check ride.
Hope it helped!