Sounds like a cable was inadvertently disturbed.  With power off, reseat the front panel connector to the motherboard.  Maybe reseat the power supply connections, too.

Thanks for the suggestions - have tried your thoughts.

There seem to be only two power cables from the power supply to the motherboard - P1 and P2, and they are both solidly fixed.
The only (relevant?) cable on the front panel is the power switch - and the cable from that to the motherboard seems fine as well. I haven't yet taken that connector out to check power switch as it is difficult to get to.

None of these cables go anywhere near the battery - so think it unlikely they have been disturbed.  Would any other connectors stop the system reacting - there isn't a master fuse somewhere?

Not sure if relevant but I have the SFF 790.

If I cannot fix I think I will go ahead and buy another of the same model - and just swap the drive.

Any help appreciated

Check the chassis intrusion switch and cable that connects to motherboard.

Flashing Amber power button is bad power supply.

When you changed the battery, did you short the BIOS reset jumper for at least 15 seconds? If not the the BIOS settings could have been seriously mangled by acute undervoltage ... the reset jumper discharges all the power in the CMOS & so gets you to a known "factory" state.

If you can't find the reset jumper, leave the battery out for 3 hours to give the CMOS a chance to run down its picoscopic power reserve. (The very low current consumption is the reason these batteries last at least 5 and more typically 8-10 years in normal use)

Thanks for the suggestions - pressing the button on the BIST causes the LED to light up - even with everything plugged in - which should mean that PSU and MB are good I think.

Have also checked the power button - it does work, at least closes contact when pushed according to continuity meter.

So I am left with a problem - pressing the power button still seems to do nothing - no lights, no fans, no noise.

To restate - the PC was working perfectly until I switched it off.  Only thing I have done is replace the cmos battery.
I have tried leaving the battery out for an extended time to reset the system to default - no effect.

One other note - if I press and hold the BIST button I can hear a fan start up - so something is working!

Any thoughts / suggestions as to what to try next?

Thanks

Barring just bad luck on motherboard, go back and double or triple check connections.  Reseat the 24 pin power connector to motherboard.  

Another test is to jumper Pin 14 (Green wire) to any Black ground pin in the connector and see if it powers on (connector plugged into motherboard).  If it does, the problem is still somewhere in the power button circuitry   If it doesn't power on the motherboard is most likely toast.

With nothing seeming to work I have purchased an identical PC, with the intention to simply swap over the SSD and hopefully continue where it left off.

Am waiting for delivery - in the meantime have been playing a little more.

Tried (again) taking battery out and leaving it for 4 hours.  This time, when I put it back in, the power button started flashing amber, but still no effect when pushing it.

Holding in the BIST button causes the main fan to start and the CD powers up.  I have tried putting in a disk - and it can be heard moving the CD head to read it - but then the access light goes out and nothing further happens. I also tried removing SDD, so that only CD was connected - same result.

EXCEPT
One time, when holding in the BIST button, after the CD had stopped accessing - a light on the front came on - the second from the left of the four lights. This continued to flash until eventually I got fed up and released the BIST button.
This only happened once - since them only the CD light comes on when holding BIST button, and nothing further.

As you can see I have more-or-less given up - now waiting for the new one.  Only reason for posting again is due to the rather strange response to holding in BIST light - just in case anyone has any thoughts.

A bad PSU can still whirl up the fans when test button is pushed.  You also said you had blinking amber at one point again.

It's supplying enough power for fans, and may be a CD player, but doesn't have enough to start and boot PC.

I can't remember of it was said already, but PSU going out on a 10 yr. old PC is not too unusual.

Thanks - it does seem most likely that the power supply has failed in a way that it could support running, but not starting up.
I did try disconnecting Cd and one of the two SSD's, on the grounds that the PSU might have enough power to start up without the extra load - but no joy.

Hopefully will get the replacement PC today/tomorrow which will solve the immediate problem.

Thanks again for the responses.