Funny... when you are finally able to say it!!
I´m not sure if I got the point of what your plan is (sorry, but I´m Spanish and my English is all the precarious it can be). I see two possible interpretations of it and I may have an answer for one of them:
If The idea is getting your complete JAR license with that instructor, then yes, your flight time will be recognized anywhere. Something like that was what I did. I got my Spanish JAR PPL here and went with it to Canada. The plan was getting a validation there, and the only two things they required me to do was showing my actual license and medical. It means that they accepted that I had enough flight time to give me that validation and that they didn´t care where I flew it. In case of getting a complete Canadian license, they´d have asked me to write the tests and have the checkride, but again, no need to fly all the needed hours there. (A couple of friends got their Canadian comercial licenses starting from their South African´s, and they wasn´t asked to do it either).
But, if the idea is getting all your trainning in the JAA enviroment and then write and "fly" the tests... I´m not that sure. I´d bet they ask you to drive your trainning according to their own rules, what would mean FAA instructors and N- registered airplanes. Not sure anyway, just that, betting.
In case the original plan was the second one, you could always think about re-planning it and make it according to the first I guess. Don´t you?
I hope it helps in any way... more than confusing you!
José