You are correct - JAA PPL training must be carried out by a JAA instructor / school.
You could still have extra lessons at an FAA school and log the hours but they will not count towards PPL minimums and you will need a training Visa (ignore the people who try and convince you that because you are not training for an FAA PPL you are somehow excused. The TSA do not see it that way).
I can see pros and cons to this. On the one hand you can have some cheap lessons in good weather. On the other hand, it could be counter-productive if you are flitting between instructors and the different environment of UK vs USA.