Well that didn't take too long to fix!
I now have version 9.4.49 running quite happily on XP, and the only thing you have to do to make it communicate with the Dropbox servers is run the executable in compatibility mode for Windows 2000!
So, it looks as if it looks specifically for XP and blocks it, but an even older OS is perfectly acceptable!
Go figure!
I found that with a bit of research of course, I didn't find the answer all by myself!
It won't let you in with an old version of the client, so what I had to do was uninstall my version 2.10.52.0, and then find a copy of 3.18.1, which may well be the last that will install on XP from my researches. It installed fine, but still wouldn't let me in because it still said it was an old version, but then, lo and behold, it updated itself to 9.4.49 (the current version) and just carried on working.
So, normal Dropbox service resumed, at least for the moment!
As always, there's no way of knowing how long this will last for, but I'm very pleased that it's OK again for the moment!