Thanks for all the advice. It really was appreciated.
On Thursday evening I spoke to a local computer engineer. He said it was almost certainly a PSU failure as a result of both the age of the unit and a build-up of dust in the fans, as a couple of you suggested. He didn't think there was likely to be any other damage, so I took the PC over to his workshop yesterday afternoon. He contacted me a few hours later to say he'd installed a new PSU, cleaned out all the dust and the machine was booting up just fine.
I collected it this morning, it's working perfectly now (better than it has in years, in fact) and I was charged just 40.00! Excellent service and Neil is a really nice guy to deal with. Can't recommend him and his company highly enough. If you live in the Cardiff area and have any PC or laptop issues, his website is here:
https://wewillfixyourpc.co.uk/
I quickly managed to back up the data I thought might have been lost and now have plenty of breathing space to explore the market for some new equipment. 40.00 well spent I reckon.