Interesting but a little more complicated then you might think.
All of the JAA schools in the US do their training on N-registered aircraft in US airspace. This means the aircraft should be insured; I am curious as to the opinion of a US insurance company concerning flight training taking place by a non-US certificated Instructor.
A JAA PPL student in the US is a bit of a hybrid.
In order to solo they need a US Medical and student pilot certificate.
So for their solo they need a US certificated instructor signing their medical and logbook.
For every solo XC they need a US certificated instructor signing their logbook endorsement.
There is another problem, in order to be PIC of a N-registered aircraft you need to have a US certificate.
Your primary student can not be PIC since they don't know how to fly yet and on a student pilot certificate they cannot carry passengers. Which is what you will be in the eyes of the FAA because you do not hold a US instructor certificate. So how are you going to fly with a primary student?
You could only train a JAA CPL applicant that holds a US PPL. They are PIC and you are a passenger training them. But this is where the insurance company comes in if something happens. It doesn't even have to be anybodies fault in particular, you could blow a tire on landing and end up in a ditch with a bent aircraft.
I would get my US certificates if I were you.