I contacted our IT Services and they said that MailChimp wasn't blacklisted. I contacted MailChimp and they said that there must be some kind of filter which was blocking it. The answer in the end was to create a non .le.ac.uk address for the mail to be seen to be coming from so as to get through the spam filter which blocks messages that seem to be pretending to be from .le.ac.uk addresses but are coming from an external service. So I created a sdzhelpdesk@gmail address and set up an auto-forward on that to send all mail to firstname.lastname@example.org. I then made sdzhelpdesk@gmail the default "Reply-to" email in MailChimp and it works! Confirmations from .le.ac.uk address are now received and replies to messages are routed through gmail to .le.ac.uk.
I've tested it with three different people but if you would be willing to check for me (if you have a .le.ac.uk address) you can do so here. And as I said in my tweet, thanks to Matt, MailChimp and ITS for helping solve the problem. And to Vic for spotting it.
Now I can get on to the serious business of using it for student engagement.