It was a regular event in the past years for our UK CAA inspectors going to the US to inspect and issue EASA approvals. Its been my understanding that the EASA stopped that process stating that an individual member state could not give an approval to a 'third country'. They subsequently inspected and approved ATOs themselves. I don't know whether EASA created a department or a staff role to do this or subcontracted out as and when required.
It is true that , in normal circumstances third country ATOs have to apply direct to EASA. The department in Cologne responsible for third country approvals is the
Flight Crew Licensing Organisation Approvals Section (FCLOA) who sub-contract day-to-day oversight to individual member states. However, since the UK is
currently a member of EASA and not a third country, a number of member states have, with EASA's consent, expressed their willingness to approve established UK ATOs ahead of 31 Dec. For example, I know of one ATO that has an office in Brussels and is approved by the Belgian DGTA and another that is approved by Austro Control. Both do most (but not all) of their training in the UK but are approved as, respectively, Belgian and Austrian ATOs with an overseas element to their operation. Much in the same way that Naples Air Center, having had their own approval revoked by EASA, are, for now, approved by the UK CAA under the Andrewsfield Aviation ATO. The Belgian and Austrian authorities have no more difficulty in conducting inspections in the UK than the CAA used to have in conducting inspections in the USA, Canada, Australia, Brunei, Spain, Norway, etc.
UK CAA inspectors used to go to the US and elsewhere to inspect and issue JAA approvals and, later, EASA approvals on behalf of FCLOA. Unfortunately, they made such a hash of the latter that the contract was withdrawn some years ago and oversight responsibility was shared between other EU member states.