I'm sure it'll come as no surprise to know that it's all my fault 
The Admin account was set to forward e-mail to my work account. I never actually logged-in to that mailbox.
Unfortunately, the mail service is set up so that it deletes any accounts that haven't been logged-in after four months.
So, the Admin mailbox got deleted. I suspect that, since the server sends e-mails FROM the Admin mailbox, that's why nothing was getting out.
I've recreated the Admin mailbox and now set my Outlook client to actually check it, so HOPEFULLY, we're all sorted.
Very sorry for any inconvenience.
Steve.