When conducting a test, the Examiner is acting on behalf of the CAA who issue the authorisation to a specific person. It has nothing whatsoever to do with the FTO/RF, who are acting as a third party in recommending an examiner.
Irrelevant.
Under the sale of goods act in the UK (feel free to look it up) the person doing the selling forms a contract with the person doing the buying. The contract cannot be abrogated to a third party in the UK without the buyers consent.
If Joe Bloggs pays Fred Carno Flying School Ltd for a flight examination, his contract is with Fred Carno Flying School Ltd and they are responsible for providing said goods and services.
If Joe Bloggs has entered into a private arrangement with Holden MaGroin Esq (Flight Examiner par excellence), then said HM is responsible for providing the goods and services.
It's not rocket science, it's not negotiable and the CAA/Flying Schools do not have a remit to do any thing different.
If the OP paid the FTO, they have the buck. If the OP entered a private arrangement with the FE, the FE has the buck.
It's really ****ing simple...why don't you get it??