As, according to their website, HP clearly never made any DeskJet 840C colour
printers I must have manufactured my own whilst under the influence of some legal, or possibly illegal, mood modifiers. Or both together, of course.
No wonder getting a driver for a non-existent printer is so difficult!
But I have, in part, succeeded. And the part that hasn't [yet] succeeded? I can at the moment only print in black and white as the colour cartridge doesn't seem to be getting the message, but that's a problem for resolution later as at least I can print something now.
I can now print from both the Snow Leopard and Catalina iMacs using the driver OSXHPInkjetInstaller from the HP940 series of printers at
https://support.hp.com/us-en/drivers/selfservice/HP-Deskjet-940c-Printer-series/35985 and selecting to MacOSX 10.4 version.
Mind you, as seemingly the only person in the world to have a DeskJet 840C, at least according to the experts at HP, this information won't help anyone else - unless you too made your own 840C printer without realising it!
Thanks for the advice, one and all, yet again FineTunedMac comes up with the goods!