Under 61.75, the FAA can "validate" your foreign, ICAO-compliant PPL. You will be issued a "piggyback" (formally called 'based on") FAA PPL, which is only valid as long as the underlying PPL is valid, and implicitly carries all restrictions and limitations of the underlying PPL. (An EASA PPL is ICAO compliant, so can be used for this.)
Start here:
Airmen Certification - Verify the Authenticity of a Foreign License, Rating, or Medical Certification
To make the "piggyback" licence legal, there's two more things required:
1. A valid medical. Either the medical that goes with the underlying licence, or an FAA class III medical is required.
2. A BFR no more than two years ago. This consists of an hour groundschool and an hour in the air and will be handled by your FBO.
An M-1 visa and TSA clearance is NOT required if all you are doing is renting an aircraft and a BFR.
Note that the process may indeed take long, since the FAA has to contact your local *AA to verify the authenticity of your licence. And since certain countries, including the UK, have strict data protection laws, the FAA will not be able to verify the authenticity of your licence with the UK CAA unless you sign a waiver for the CAA that they may release your details to the FAA.