Neither the CAA nor the training organisation bears any legal responsibility in the case quoted.
Thas not quite right in relation to the flying school. Forget the legislative aviation requirement for a moment. A verbal contract has been entered into here by the student and the flying club for the provision of a service, unless of course the examiner just happened to be passing and popped in and said, "anyone fancy a skills test"?
The supply of goods and services act 1982 says that a person providing a service must do so with reasonable care. Clearly in this case a service has not been provided at all. If the flying school arranged the test (in any way)and have not stated to the student that they only acted as agent in the matter they (IMHO) are liable. If the student made all his own arrangements solely with the examiner then the examiner is liable. I would say that if the school charged the dual rate for the hire of the aircraft during the test that could prove they were complicit in the matter.