Actually, I couldn't be bothered to wait. I followed the instructions for how to open the case, from the RapidRepair link above (actually quite easy - you don't need any special tools other than a coupld of very small flat- and phillips-head screwdrivers such as you'd find in any "jewellers" screwdriver set).
Disconnected the battery, reconnected it and the device boots up fine. For now at least - who knows what will happen at midnight ;-)