I called in to the space on the way home from work just now and brought a PC tower that Tim Harbour was getting rid of a while back, which has no hard disk or RAM, but does have a Windows XP Pro license sticker on the back. As it turns out, I didn't need it.
I had a look at the current PC and there doesn't seem to be anything wrong with the switch, but this is cabled to a small header on the motherboard, with a sort of pin adapter in between. This looks easy to dislodge and I'm hoping that is all it was — a poor connection. The switch doesn't switch power to the motherboard per se, as it has this all the time, as indicated by the permanently lit green LED on it. Instead the switch just shorts out two pins to signal to turn the PC on/off.
I tested by turning the computer on/off about 15-20 times and it worked fine. So hopefully that's it. if it proves problematic again, try removing the power lead for a minute or so, replace and try again. This seemed to work when I was experiencing issues previously and so it may be that this helped and an issue remains — perhaps with the power supply or motherboard — but hopefully it was just coincidence and/or removing and inserting the mains lead shoogled the innards about enough to make the power switch connection work.
Only time will tell.
In the meantime we should look to find another PC, which doesn't need to be as powerful, to replace this one with. Then we can re-purpose it for something less critical, but where it's better used, e.g. to run CAD.
Much as an XP computer would technically suffice, it would be far from ideal given that it's no longer getting updated, so more prone to viruses etc. So we'd need a Windows 7 or better machine.
Remember: if the current PC stops working, remove the mains lead, wait a minute and re-connect it.