Well, it's confirmed - the rear LCD is decidedly, definitely dead.
I got the mini-screwdriver out, whipped the back off and took the lcd out, cleaned the connectors hoping it was just a dirty contact, but nope. On the plus side, it's really easy to get in and out, so it's just a case of buying a new one and sticking it in...

The other good news is that the clock battery works. I woke up this morning and suddenly remembered that there's a video output - so I plugged the camera into the telly and set the date and time that way! I also did a few other things like turning the image preview off, no point if the screen isn't working - might make the battery last a bit longer (the screen backlight still works).