Ok, so the good news first. This was the outcome of yesterday's M.O.T. test:

After me last (incredibly naive) post, I put the Z on axle stands and discovered that the offside trailing arm looks like this:

Before I bought the Z I did check out the M.O.T. history and it had failed for a rusty trailing arm a month or so before I bought it and a week later it passed, so I assumed that it had been fixed. I did have a look underneath when I bought it and at least superficially it looked much better than the other one, but I do wonder now about how legit that last M.O.T. test really was.
For a brief moment I wondered if it's worth fixing or if I should just sell it for parts. Then I found out that it wasn't the first time in the Z's history asked the same question:
Hi guys thanks for addind to the group i have had my Z3 for 10 years now and i have just had the MOT done and it is failed on the rear off side Trailing Arm and i need to replace it but having difficulty in sourcing one any of you good people any ideas as to where i can get one from maybe best...
zroadster.org
I can confirm that he sold it...
Anyway, I did fix it because I knew I would have to buy another one if I didn't...
So for about 3 months, the drive looked a bit like this:


For a while, the trailing arms hung from the ceiling in the spare bedroom while being painted with Hammerite:

And because it seemed to be always raining, the living room looked like this while I installed the trailing arm bushes and put the arms back onto the beam:

But we did it in the end. I had lots of help from my neighbour who is an engineer, BMW fanboy with a well equipped workshop. And
@Lee who very kindly swapped the beam bushes for me. Again, many thanks mate!
Obviously not everything went smoothly. We spent a whole Sunday afternoon with removing the "bolts of doom" from the beam plates. We got 3 of them out in one piece, but the fourth had to be drilled out. The nearside driveshaft was impossible to remove from the hub. In the end I gave up and ordered a new one. But the most frustrating thing was that when it was time to start putting it together again, I said to attach the diff to the beam first, then jack it in place and attach the beam. But matey was adamant that it was no problem to put the diff up separately, because "I've swapped the diff on my E30 loads of times mate". Well, we managed to cross thread both the pretty much inaccessible upper bolts that hold the diff to the beam, which caused quite a bit of swearing and throwing spanners on the ground in frustration. But it's done and I'm rather quite that we managed to do quite a big job successfully. And as the sun's come out, I'm gonna go now put the roof down and go for a little drive...