Law says that tyre tread, must not protrude the wheelarch...
However, what passes an MOT, and what a Police officer deems fit to be on the road, are 2 entirely different things, so keep all things within reason, and make sure you have arch clearance, with zero scrubbing especially when fully laden!
So it depends on who's looking at your car, and whether or not they have had a good day!
Thom