Spoke too soon

Turned out there was a problem with BINC IMAP. For some unfathomable reason it wasn’t checking new mail, i.e. the files in the‘new’ subdirectory of each maildir.

I downloaded Dovecot and installed that. Everything’s working perfectly now, I just deleted some messages on the ThinkPad using Mozilla and saw them vanish from Apple Mail moments later. Thanks to Clive for the recommendation.