Thanks again Owl-53, also thanks to HWTech. I sometimes wonder what I would do without the Apple Community. You guys are all great!
To elaborate on my previous post: This whole debacle started with the sudden and earlier than anticipated installation of our new Starlink Dish and Router (which seems to be working very well as far as I can tell). In changing over from one ISP and WiFi to the other I had a few problems to solve at once: getting our WiFi printers reconnected (eventually solved): Getting Cyberduck (FTP app) to work again (not completed but tolerably useful); and trying to get Mail working properly on the Sequoia Mac. It was this last issue that led me to discover that the OS hadn't and couldn't be updated.
After the reply from den.thed my investigation led me to use Terminal to download the Tahoe installer but the Installer failed to complete downloading the OS. I repeated this, but for Sequoia, with the idea of updating to the latest version but encountered a similar problem. The only thing I tried that I believe would have actually changed the OS or any other interfering Apps in any way was to run Disk Utility to do a Disk Repair on the Hard drive. There didn't seem to be any issues with the disk but this still did not result in showing Tahoe was available. I had rebooted the iMac several times during all this.
Note: There is no anti-virus software, cleaning & optimizer apps, and third party security software installed on this machine.
I did not nuke the Hard drive as considered because I discovered that I didn't have a recent back up to restore it with (failed Time Machine disk). After installing a new Time Machine drive and letting it complete its first backup, I decided to work on the aforementioned Mail issue, matching the account details with a MacBook that was working. During all of this I was also testing and checking my email accounts were all working ok on my own iMac since the ISP change (still having a bit of an issue with gmail!).
It was after 11pm, after I was reasonably happy that most major issues had been resolved, I decided to check if Tahoe was available on the iMac in question before I went to bed, and lo and behold, it was! I started the installer downloading Tahoe and went to bed. This morning I entered the required password to install Tahoe which it completed without any problem. Had this not occurred my next step would have been to nuke the Hard Drive.
I appreciate that you guys may have never encountered the OS updater being stopped by a failed Time Machine issue but I believe that I did nothing to this iMac during all this process that would have had an effect on the OS. Restoring Time Machine was the ONLY thing I successfully managed to do. Our 2 MacBooks do not have active Time Machines and updated without a problem. Hence, my theory. Perhaps it is merely a coincidence, but I elaborate all this only with the intention of hopefully helping some other frustrated individual with a similar problem.