I've had some success today.
Thanks to
@bkh I decided to try safe mode.
- I booted into safe mode and then into Disk Utility.
- I ran Disk First Aid on both my Macintosh HD and my External Time Machine USB Drive.
- I then started a Time Machine backup whilst in safe mode and it completed successfully. The files were visible in Finder too.
- Finally, I restarted my Mac and initiated another Time Machine backup which completed without any problem and the files were still visible in Finder.
So far so good...
I am now trying to do a Time Machine backup to my Synology NAS. It says "about 4 hours remaining" so I'll update this post when it has finished.
Hopefully this will help some of you with your TM problems.
**UPDATE**
The Time Machine backup to my NAS gave the same "Waiting to Complete First Backup" error on completion.
I then booted into safe mode again and started another backup to my NAS. That completed quickly and successfuly.
I restarted my Mac and did another backup to my NAS which also competed successfully.
I have no idea what is causing the problem but, for now, it seems to be working.
I'll report back if I have any further problems.