Technically you should then tax it from the start of the month in which you rode it.
A situation could occur although the chance is slim that you could be captured by one of the DVLA's mobile roadside cameras,this would immediately generate a computer produced letter untouched by human hand to the registered keeper informing them the machine was sorned and had been seen on the road, this would also contain a claim for all the revenue owed since according to their records the machine was last taxed and a fine for using it on the road!
No amount of arguing that it was booked for a pre arranged test would stop them pursing you for the amount owed ,(they've heard it all before).
Their revenue has dropped considerably since the abolition of the tax disc, they use very strong enforcement to recoup revenue from evaders.
Simple solution, get the mot done and tax it from the first of that month.
You can return from MOT and put it back in your garage, still SORNed. No need to tax it until you wish to ride it other than covered by the exemption.
I wouldn't get overly worried.
You are exempt from needing Vehicle Excise Duty (road tax) if travelling to or from a pre-arranged MOT (or SVA initial inspection), or on failing the test, to a place of repair. This MOT failure will of course be on the test centres (DVSAs) computer, you'll have a certificate, so it is provable later. As long as that was your journey, and it's currently SORNed, you'll be OK.
Obviously the DVLA camera van will not know the purpose of your journey, so the computer will likely send you a penalty notice, but a simple reply with a copy of the pass/fail certificate will prove your exemption.
Same exemption for the expired MOT. You're more likely to get pinged by a Police car fitted with number plate reading cameras, who will either stop you immediately, or totally ignore you if they're already heading to another job, except if the screen also shows No Insurance details held. After stopping you the officer may phone the MOT test centre to check if you have a test booked, as the exemption legally only applies to a pre-arranged test, not a 'just turn up' test.
Links to legislation if you have insomnia
For the exemption to road tax when going to/from pre-arranged MOT
see Schedule 2 Section 22(1)(a)+(b) of the Vehicle Excise and Registration Act 1994
The DVLA normally show no discretion (they have literally heard it all before) when dealing with owners who fail to SORN/tax or forget to notify change of keeper, but in cases such as this, there is a legal exemption, but you'd have to show them you're exempt.