However the JAA [sic] flight schools in the US operate under the oversight of the UK CAA
Flight schools that have their principle place of business in the US (or any non-EU state) are now regulated directly by EASA. Oversight is contracted out to member states and the UK, for the time being, is providing oversight of most US-based schools. The word coming out of Cologne, however, is that EASA is less than happy with the resources that the UK has available to fulfil the contract and may well look to change contractor in the future. Schools that have their main place of business in the UK but conduct partial training outside the EU (e.g. OAA, CTC) remain the responsibility of the UK CAA