When I was there in 2013, it was still supervised by UK CAA and I imagine it still must be?
The competent authority for all 3rd country approved training organisations is now EASA., although the day-to-day oversight is sub-contracted to an individual member state. This means that whilst an individual ATO may be 'supervised' by the UK CAA (or another competent authority), the approval certificate is issued by EASA.
FCL.025 states, "Applicants shall take the entire set of examinations for a specific licence or rating
under the responsibility of one Member State" - note that it doesn't refer to a single competent authority. In practice, this means that, provided there is a 'Ground Examiner' at El Cajon, you may sit UK PPL theoretical knowledge examinations there notwithstanding that the UK is not the competent authority for the ATO. However, as the ATO is subject to the provisions of Part-FCL, they will need to provide the minimum approved classroom training and make a formal recommendation for the particular subject(s).