In case anyone else is having - or has had - similar problems, here's how I (..so far, anyway..) fixed it:
I found - by looking things up online - that the little 'CMOS battery' is there just, apparently, to maintain the date and time when the Mac's off. But my date & time weren't failing; they were always OK ..so: not the tiny, almost inaccessible 'CMOS battery'.
I decided that this appeared to be a breakdown between external devices (mouse, keyboard, wi-fi and bluetooth systems) and the DISPLAY ..that's to say graphics cards.
iFixit's demo of how to reach the 'CMOS battery' shows - on the way to the battery - the removal of the connections between the interconnect Board at the very bottom of the trashcan Mac Pro and the graphics cards. The connection is a flat multi-pin cable at the very bottom of the machine, but easily accessible as soon as the cover's been lifted off. I thought that perhaps this connection may have been insufficiently tight (..it may have been disturbed by moving the Mac, or general knocks and bangs, or just not made tight enough at original assembly).
I also decided to re-upgrade the OS software BY ETHERNET CABLE, instead of trying to use wi-fi (which disconnected after 25 minutes).
So I opened the case - it took 5 seconds to lift off the cover - then prised off the graphics cards' connections (the flat black connectors at the base of the graphics cards), then FIRMLY re-attached those multi-pin connector cables, put back the cover, plugged in an Ethernet cable, waited for 5 seconds - after re-inserting the mains cable - before re-booting (to reset the power management system), restarted holding down command+alt+P+R to reset Parameter RAM, then shut down, then restarted with cmd+R in 'Restore' mode, and re-downloaded the latest (10.15.2) 'Catalina' OS.
The machine ran perfectly during download and installation and then restarted ..and has now run perfectly ever since, including sleep, wake, power down, restart, etcetera, and seems PERFECTLY stable.
It appears that either - or both - FIRMLY reconnecting the graphics cards plus re-installing the latest OS has fixed my - and perhaps your - problem.
Thanks to iFixit Tech Support for clarifying exactly what the 'CMOS battery' does (and doesn't) do. I haven't found any other documentation which explains what this problem might be, or how to fix it, so I hope this helps!
