For insurance reasons there are two things to consider.
Having no MOT... this Does NOT automatically invalidate your insurance.
Driving a non-roadworthy vehicle... this can invalidate your insurance.
So there is a grey area there... for example, my car could fail it's MOT due to defective windscreen wipers, but this wouldn't necessarily make my car unroadworthy on a dry day, but if it was peeing down with rain and I had an accident it could be shown to make my car unroadworthy at the time of the accident, which could invalidate my insurance.
I've also tried to find a definitive answer to "does a MOT failure invalidate an existing MOT" over the last few weeks... and I can't find one, even looking at official sites there are contradictions.