The new RAM stick arrived today. And did nothing. It's not the RAM.
Got the multimeter out and checked voltages coming out of the PSU: everything perfect, all within correct ranges, exactly as it should be.
My only conclusion now is the motherboard has gone. Power is running through it (fans are spinning, an LED on the PCIe USB card lights up), but somewhere there is a fault preventing it working. I can't think of anything else. And there isn't a single speck of dust inside the computer now either, all fully cleaned just in case dust was causing a short. All add-on components (graphics card, etc.) have been removed, cleaned and put back. Cables have been removed and re-attached.
I did reset the CMOS (again!) because... well, you never know.
It has to be the motherboard.
Unless anyone else here has any ideas? They would be very gratefully received as I am causing a bald patch with all the head scratching.