Is there a lot of tread difference front to back.
Any more than 2mm i think it is will give problems.
As said do the chalk test before it does any more damage.

3.0d M57 ? or N my bet is the m.

What engine is it or did i miss that bit.

Black smoke = too much fuel ??.
I would still try a leak off test.

I had the 4B90 on mine with the m57 it turned out to be 2 injectors leaking.
My car ran ok once started not bad to start from cold but a bugger to start hen hot.

Result

What fault codes come up on a code scan.

The adjuster for the shoes is accessed through a wheel bolt whole and better with the wheel off.

Don't see how with today's MOT kit it finds the car on the system.
Even the brake tester is live to the DVSA now.
The only bit i see where you can cheat a bit is where it asks for the plate value for emissions.

Asked 2 testers at my m8's garage yesterday and both gave the same answer. Testers deaccession and they have tested a few of the older E53's with no EML light and couldn't remember ever seeing one. I never had a problem and my sister has just well last year got an 2002 E53 again with no EML light an...

Send it back asap.
Don't no much about them but i know of one that is forever in the garage.
The last thing i remember failing on it was the brake servo and for some strange reason they put that inside the bell-housing and what a job.

Another thing is they should not run diagnostics during an MOT.
They plug the MOT computer dongle in so the computer can see the engine temp and also the rev's when doing the emissions test.

My E53 has no check engine light any faults will show in the screen.
Same as glow plug light not there but says "engine pre heating" ell something like that.
I know i have seen this on here a few years back but can't find it.

That was the first link i found not the best price.

I use some of this but not for the wax but just to get the water off the car. I have the well what they call a large towel 90x60 i think it is and it will dry the full car no problem. https://www.ebay.co.uk/itm/404305411676?itmmeta=01HX3GA30X89Z7DJHVXCCM4HQG&hash=item5e227afe5c:g:sDQAAOSwEKBkb2u...