Hi all.
Bad news, and much better news:
Sgunhouse: The bad news is for my business account, I had checked that box, so they were all downloaded and stored locally. But when I hit on the bright idea of fixing the original problem by reinstalling the account, they all got deleted, and then, as per original message above, only the headers downloaded! (Lesson learnt!)
The much better news is that new emails started coming through properly this morning. So I deleted the account again and reinstalled it, and now they're all back as they should be - with correct header, date and time details, AND all the content downloaded too.
No idea what the problem was, but it is all resolved. Thanks to everyone who read my message, gave it any thought and/or replied - much appreciated!