If you are driving to a pre-booked MOT test you must be insured. Your insurance, under these circumstances, will NOT be invalid.
I've looked for a definitive answer for the rest of the time but it seems to be a grey area. From Parkers:
Driving without a valid MOT
Driving without an MOT won't always make your insurance invalid. In most instances the insurers will still pay out in full - and if your car is stolen, or damaged, the payout will usually only be reduced to reflect the value of the car without a current MOT.
However, some policies state in the small print that an MOT must be in force. Only in instances where the vehicle has a fault which contributed to, or caused the accident, can the claim be rejected. Driving without an MOT does carry a potentially hefty fine though.