The way I read it is when you tax your vehicle you are asked if you want to declare it as a VHI (Vehicle of Historic Interest) which gives the vehicle MOT exemption. But if the MOT is up before the tax then you don't need to MOT but you must declare as a VHI when the tax is due, if you don't then you need an MOT before legally driving the vehicle on the road.
I always thought that the vehicle had to be MOT'd before you could claim VHI but it looks like I was wrong. Bonnie's MOT and tax was due last month but there was no declaration when I went on the DVLA site then. I did ask my MOT man about it and he was unaware of the 40 year exemption but was well clued up on all the other MOT updates.
From the DVLA website
https://www.gov.uk/government/news/mot-changes-20-may-2018