There has been a lot of chat on Facebook Droopsnoot group. Which lead me to clarify their position at DVSA
The confusion of what you nee to do to go MoT exempt is starting to become cleared - the answer is
nothing!! I clarified with DVSA this morning that their default position on verhicles 40yrs old and older is THEY DONT NEED AN MOT! So if you want to tax your car and your MoT has run out then you can. This would appear to apply to modified cars which are still on a year related registration. If you want an MoT you can as the MOT station to MOT it and put the results on the system. If it fails you need to fix it prior to it becoming legal to drive within the rules of the failure category. You can ask for an informal check against the MoT check list which is what I have done with Alfie. I have taxed 2 cars after the MoT had expired and both were confirmed as taxed for a year without doing anything else that follow the DVLA on line tax process as I have always done - no boxes to tick for applying for MoT exempt - just the standard process we used before the May rule change.
Don't ask me about 1978 vehicles which are not quite Tax exempt but could be MoT exempt? That would seem to be another bag of worms. Need someone with an expires MoT on a 1978 car to try and Tax it and see what happens