New tyres what if

Monday I took my van for its mot , its all managed through a fleet management company, the van passed, but I was told that the two front tyres were at 3mm, but when the garage asked the fleet company if they could change them they were told they have to be 2mm minimum before they will change them, fair enough, Tuesday I noticed one of the rear tyres looked a little low and after inflating it I noticed I had picked up a nail in the tyre, off I go to Kwik fit and they couldn't repair the tyre so a new one was needed, the fitter also said my two front tyres were at 2mm so they were authorised to change them, now the mot garage ( main dealer) I assume had measured them incorrectly, so that got me thinking, if the garage who did the mot had said the tyres were say at 2mm but were actually worn down below legal limits, and I were stopped by the police, who then checked my tyres and decided to issue a fixed penalty notice, how would I stand if I had a valid mot , done the day previously with all the documentation?

Re: New tyres what if

Tyres have markers that show when they are worn out/illegal surely, I just go by those, no opinion needed. When they are level with the tread the tyre needs changing. Presumably the police do the same thing.