iCloud Mail sent items stopped syncing between devices

I have an iCloud mail account which is set up in Mail on my MacBook Pro, iPad and iPhone and it has been working perfectly well for ages. Suddenly, in recent weeks, messages sent through this account are no longer syncing between devices. Messages appear in the Sent folder on the device from which they have been sent and on the iCloud.com web portal, but they are not then downloaded onto the other devices.

I have searched high and low for solutions and have tried all the obvious ones: restarting devices, disabling and re-enabling the mailbox, etc. The Mailbox Behaviour settings are all correct (sent mail set to save in the Sent folder on the server – which, as outlined above, it does).

What is interesting, is that if I look at the sent folder on, say, my iPhone, and refresh the view, it will occasionally say "downloading x messages", so it recognises that the messages are there on the server, but then nothing happens.

This is extremely frustrating – can anybody offer any suggestions on other things to try to get it working again as before?

In case anybody wants to know, all devices are running the latest release versions of Mac OS/iOS/iPad OS.