Continuing the saga of yesterday, I took the new machine back to Hard Drives Northwest early this afternoon where they very helpfully plugged the machine and watched it boot fine. They were very helpful and suggested that it might be a power problem - I had been powering the machine through a UPS.
So I went back home, plugged the machine directly into the wall and watched it fail to POST again.
Just a flashing cursor.
Then I had a little brainwave. It's plugged into a 24“ monitor. Could that be it? Surely not - it's connected via VGA.
Unconnect video cable. Power on. Wait 10 seconds. Plug in video cable. Success!
Being an ex-Graphics guys I've a ton of old video cards lying around, so I plugged in a spare ATI PCI Express card, plugged the monitor into that and I'm off to the races! The machine at least POSTs now and I can get into the BIOS.
In summary (and to capture this post in search results for others to find), connecting a Dell 2405FPW 24” monitor into the onboard Intel Extreme Graphics of an Asus P5K-VM motherboard appears to cause the motherboard to refuse to POST.
Next up - install!