Finally got a chance to take another look today.
Scrabbling around under the dash, I discovered a blown 30A blade fuse - not in a proper fuse block, just plugged into a couple of spade connectors.
I didn't have a 30A fuse to hand, so I borrowed a 25A from my Land Rover.
Before I plugged it in, I put a meter across the connection, put her in 4th and turned the ignition on. Flicked the OD switch and... Eureka! 12V (ish).
This was looking promising. I plugged in the fuse and took her for a drive.
Sadly, still no overdrive.
When I got back, the fuse was still intact. I could also hear the relay clicking on and off.
I'll test the voltage from the relay tomorrow - if I can get to it...
But it's looking like the solenoid might be fried. Does that sound about right?