Tom,
Many thanks for your valiant efforts but I've given up and reinstalled Windows. (Actually quite painless for me since I've got decent backups and have local copies of all the bits I need. I'm typing this now on the rebuilt OS.)
Your last set of instructions didn't work for me. Trying to replace the owner globally gave me an error (despite running in Adminstrator mode), and TrustedInstaller didn't exist as a user in the list :-(
I assume that, since we were fixing the symptoms without necessarily understanding exactly how Windows Update got in this state, there was always a chance (esp. by resorting to such global CBS reg changes) that we'd leave things in a slightly inconsistent state (e.g. with missed updates).
Anyway, many thanks again, and I learnt a few things in the process. I'll change the post title to indicate [CLOSED] (rather than [FIXED]); feel free to change it yourself if you prefer another convention.
If you ever need any help with your maths (or any programming/software design stuff, esp. if it's Java or Python-based), just email/PM me since I owe you one. (Though my maths is *very* old and rusty since my maths degree is from 1995!)