Where are you going to fly the plane? The FAA will allow you to fly a N registered aircraft in Australia on your Australian license. If it is going to be outside Oz, then yes, you will need a FAA certificate.
The ATP is only required for air carrier operations. To be paid to fly a privately operated aircraft, you only need a commercial certificate. In order to log PIC time, since all turbo jet aircraft require a type rating, you'd need a type rating in that type A/C.
In order to get a FAA commercial certificate, you must hold a FAA private certificate. The FAA will only grant you a PVT based on your JAA license. Here's where it gets tricky. You can can a FAA private MEL with just a lot of paper shuffling. Or you can just get a normal FAA private cert., written test, flight training and checkride. To add instrument privileges to a "based on" cert. you can either just pass a written (knowledge) test or complete the normal training program.
If you use the "based on" PVT to get your COMM, whatever license it is based on has to be current when you apply for the COMM. The FAA guidelines say if you complete the "normal" instrument training it is good when you upgrade your certificates. I don't see anything covering what happens if you just pass the Instrument Foreign Pilot written.
I'd say you have two paths. You can let the JAA lapse and do a normal FAA PVT, INST and COMM+type rating if you're looking for the PIC time. Or you can keep the JAA valid, but I think you'd still want the do the normal INST and then you'd still have to do the normal, stand alone COMM and type.