Thanks @stevewhitemd. That has identified - as you suspected! - that iCloud is the culprit. A bit more investigation leads me to think that it is a message 'being sent' that is clogging things up; when I open Mail with all accounts enabled and online (checked via Connection Doctor), things seem to grind to a halt and bottom left shows 'Sending Message...46kb' and that stays there for hours. When I then try to, say, compose a new email I get the beachball, and then when I try to quit Mail, I get no response so have to use Force Quit. The trouble is that now we don't seem to have an Outbox any more, I cannot identify what the message is, nor how to get rid of it from the 'queue', or whatever the equivalent is. I'm too late to ask Santa for a new Mail system(!), so any ideas would be gratefully received. (And do have a very happy Christmas btw!)