You can easily get an FAA PPL "based on" your JAA/EASA PPL. It's informally called a "piggyback" PPL. Details are in 14 CFR part 61, Section 61.75:
eCFR ? Code of Federal Regulations
Note that some countries CAAs (including the UK and Australia) may not release your data to the FAA without your authorization. And the FAA will contact the CAA that issued your licence for verification. So for these countries the steps are slightly more complex, as you need to authorize your CAA to release your details to the FAA.
All details about the process are here:
Airmen Certification - Verify the Authenticity of a Foreign License, Rating, or Medical Certification
Furthermore, you need to do a BFR before the piggyback FAA PPL is valid. This is usually combined with a checkout that's a typical FBO requirement anyway.
On the other hand, why limit your hour building to the UK or go to the US? Why not make the grand tour of Europe? Even in a light aircraft you should be able to outrun/avoid most of the nasty weather if you always plan a few days ahead, and don't have any specific destinations in mind. And at most airports you'll find friendly people who don't mind helping you find a hotel and taxi, and who will help you find a spot for your aircraft overnight.