Hi folks,
I just wish to confirm that Serge Rosenthaler's solution to this problem (originally posted at https://discussions.apple.com/thread/250829294) really does work, and to thank him for it. I have no idea why Apple Support is not aware of this simple fix, or if it is aware of it, why it doesn't advise it. I haven't experienced any problems since using it.
When I was eventually able to sign in to my iTunes account, I received a message saying that I was being transferred to the correct iTunes Store. So the problem for me, it seems, was that the user database was corrupt with respect to the country setting for the store. This is not something that can be corrected by changing the time / date settings (which Apple Support repeatedly proposed), or any other setting. Even if the problem (corruption of the user database) is caused by family sharing, switching this off does not do the job. Other dead-ends included ensuring that the firewall was off, checking network settings, using different types of connectivity to the internet, etc. etc.
I strongly recommend a clean install (i.e., with prior disk erase) of Catalina for *all* devices, as otherwise syncing issues also arise. Also, prepare yourself for the fact that most of your apps (and Steam games) won't work after the update. Finally, you might (as I did) have problems with the "Find Me" app. The only solution to these (as far as I can tell) is changing your iTunes account password.
The transition to Catalina is painful.