OK, first of all: apologies to those who said it was the PSU, and how I doubted you.
Now, the long version...
As promised, after putting the two cards back in their respective machines, I tested the main rig. It was a bit slower than before, but it locked up as usual. I didn't quite follow plan A, but only replaced one of the dedicated PCI-E power leads with a molex plus converter. The problem remained. Then I added an extra fan (the case is an Antec Nine Hundred, so I fitted a 120mm fan to the outside of the case in the side door. This meant removing the perspex fan holder, turning it over and fitting the fan to the outside. Still, you've got to suffer for your art...) and tried it. Yep - it locked up again.
Next up, a bit of software cleaning. After the usual (and already done) AV, Ad-Aware and Spybot sweeps, I uninstalled all the nVidia drivers. I then installed the chipset drivers, DX9C and the last working drivers, 158.19.
Yep, it locked up again. This time it was back to locking up while I was fiddling with the graphics options - hardly a stain on the card. So finally, a probably belatedly, I swapped the PSU with an identical 650W TruePower Trio in another machine. Yep - now it works perfectly. At least as perfectly as Oblivion ever works. I managed a solid two hours play with only one overclocking-related crash (they're different).
So, it looks like it was the PSU. I'm still a little puzzled, because in my experience PSUs either work or they don't. Mostly don't, if they are Hiper ones. This one seems to be slowly deteriorating rather than just keeling over. The "duff" one works just fine in machine #3, but then that's an old socket 939 rig. It only had the Antec 650W in it because the older Hiper (of course) blew up. Soooo, time to try and get a warranty replacement....
M