I wonder why, in view of the seriousness of the accusation, the CAA didn't lift the phone to call the examiner to enquire about why he (according to their info) did a test when not authorised.
It would have found the solution long before their letter was ever written.
Indeed, assuming that the op contacted their school, the examiners reputation might have been quite damaged by this little incident, and the CAA are the only ones at fault here.
That's my view too. It must have cost the CAA a few bob to return this man's logbooks etc by a courier. They should have got onto the examiner like a ton of bricks
first for bringing the flight training system into disrepute. There are quite a few people who, having got a duff flight test, would have gone totally off their rocker and just sued, or made a big scene in front of a lot of potential customers in the school. A PPL is a huge investment in time, aggro, time, aggro, money, aggro...
However, while this has never happened to me, I know for sure that the FAA would also have washed their hands of a checkride done by a DPE whose papers have expired, or have some sort of irregularity, and they are quite happy to do so a long time after the relevant checkride.