An MOT is only officially valid for the time that it is being tested, you could pass an MOT with 1.6mm of tread on yours tyres, drive down the road and be stopped by a copper who does his test and get 1.5mm. 3 points and £2500 later you realise that your MOT certificate is worth nothing. It is YOUR job to ensure YOUR vehicle is roadworthy at all times, if it isnt and you do have an accident you can bet that the insurance company will wriggle out of paying
As per OP i am of the opinion that the new MOT overides your old one, just as it would if it had passed the test