After a while I opened it up again and it had a dialog in the centre of the screen saying 'Android is upgrading... Please unplug the power adapter before updating the Pad firmware'. I did; it turned off.
Subsequently, though, it turned on with the power and dock connected and everything was working as normal. The dock was charging, but the tablet wasn't.
I've done a factory reset from the Settings menu to see if there was something about the interrupted firmware update that had stopped it charging and it still worked afterwards, but still wouldn't charge. I did an erase/reload from the Bootloader menu and that didn't help either, but then prompted me for a system update. This insisted I had the power cord in (which seemed to make more sense...) but after the first reboot it didn't restart. I gave up at that point and, when I went back it was saying 'Android is upgrading... Please unplug the power adapter before updating the Pad firmware' again.