Driver SOS (UK)
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Instructor will not refund test fees for unusable car
I know it must be very frustrating for both you and your daughter but there is no set law in the UK to say that a driving instructor must refund the driving test fees if his/her car is deemed unusable for the test. The Driving Standards Agency states that the car to be driven for the test must be fit for that purpose and the examiner can cancel the test if the car does not meet up the requirements set out.
However it is upto the driving instructor to re-imburse the fees. Some will do it out of good faith and to maintain a good reputation while others don't usually care. If your daughter had gone with a driving school, like BSM, you would have been able to make an official complaint to the driving school and possibly get your money back. But with private driving instructors, it ain't easy and not really worth the hassle.
I'd suggest you talk to the DSA (the line for booking the driving test) and explain the situation to them. Ask them to re-schedule the test as it wasn't your daughter's fault.
That's an absolute disgrace. You should publicly humiliate this instructor. I loaned my car to a PDI for his test. My car mechanically broke down which could not be helped. It was serviced and maintained but I still paid for his retest as it was my car and I felt responsible. Doing the right thing is more important that money and people who take advantage for greed are not fit as humans.