The principle is that to fly an N-reg, you need a current FAA PPL or higher. This means having done a BFR, and having a current FAA medical. To fly a G-reg, you need a current JAA PPL,. This means that the license itself needs to be current (five-year renewal cycle for the PPL), the SEP rating needs to be current (two-year cycle) and your JAA medical needs to be current (cycle depends on your age).
In addition to this, both FAA and JAA specify that you need to have done three landings in the last 90 days if you want to fly with passengers.
The UK CAA has a blanked exception that if your FAA PPL (or higher) is valid and current, you can fly a G-reg on this, worldwide, but you are basically limited to PPL, Day VFR privileges.
And if your FAA license is somehow "based on" your JAA license (or the other way around), the whole house of cards needs to be valid and current.
By the way, you can't have both a JAA PPL and a JAA CPL, as far as I know. The CPL replaces the PPL once you passed all the CPL exams.