I just heard a rumour that the FI's at Oxford in the USA are not JAA qualified
They are not JAA flight instructors - correct. Nor do they hold JAA licences. They simply have done a little course which allows them to teach JAA syllabus.
In my opinion this is why that training in the USA is regarded as a low standard. (And Yes Oxford has some of the worst output I've ever seen IMHO)
There is only one school in the USA which uses JAA licenced pilots only. (Who have FAA Licences as well) Check their websites for details.