Thanks to everybody who told me that there wasn't a simple way to persuade Mutt to work with mailboxes containing unread messages.
I've now discovered the mutt-ng project, packaged in Debian's experimental branch. This contains the infamous mutt sidebar patch which caters to my needs fully:
- I can toggle the display of the sidebar by pressing "b".
- From there I can jump to the name of the next/previous mailbox with unread mail.
- Once on a mailbox name I can open it easily.
Job done.
Expect a more involved writeup shortly ..