I also read on other forums about people using or suggesting to use a hairdryer. And it worked easily too. The heatsink got really hot, which just goes to show how good they are at conducting heat quickly. I think had it been more than just thermal grease I would have needed to try something more, so lucky it was just that.

I've now got that system fully working again and the owner collected the PC today. It turned out to be the PSU that was faulty. I think it developed a bad ground because it was pulsing power to the system even when in soft-off mode as it was still feeding power and making the CPU fan spin and the front power led flicker, even when the system was off. I stuck a new PSU in the system and it works perfectly again.