Ewan – i think the point is if it has just failed the test and is being taken for a pre-arranged repair directly from the test centre it is fine, otherwise it is a no-no. And “repairing it at home” doesnt count – I’ve enquired with the local police as mine is currently off the road.
Regardless of whether it is currently taxed, using it without an MOT on public roads for any other reason other than a trip to or from the MOT station is illegal.
Liability is no longer yours when they sign for ownership and you havent hidden anything from them, yup. As Steve says – make sure you send off the paperwork to transfer ownership, not them. Any problems in the intervening time would fall on their liability once the confirmation comes through from the DVLA.