I swapped out the power supply but the one from the 'spare' machine didn't work at all, so I swapped in back to check that the original config still behaved in the same way. It did!
Next I swapped out the motherboard, retaining the RAM from the original, it that didn't work at all either. So I swapped it all back. Looks like my 'spare' machine is not much help.
Apart from the motherboard, the only other device connected is one hard drive. As previously, P.O.S.T. completes OK. Memory test seems OK, but then it just keeps restarting. Sometimes I can get into the BIOS setup screen before it re-boots, but often not.
I have checked the ATX power supply voltages with a multi-meter and they seem OK.
Pretty much out of ideas now. I wonder if anyone can help with some ideas?