Murdo, not sure how to contact them, but in the meantime got my nephew to sort it. Don't know what he did but need to find out for future reference.
Thanks to you and young geezer for taking the time to reply.
On the two or three occasions I've looked into a similar problem for friends, it's been the case that there's been a typo in the 'To' address(es) of the oldest email in the ones that are piling up.
Thanks 4candles, but that wasn't it. The emails were being sent to addresses I had used many times and filled in automatically.
I know Outlook is supposed to be better than my previous Hotmail, but since I changed over, it's been nothing but problems.