Once again, after an update, the NM applet refused to co-operate and started showing all connections as 'unmanaged'.
This relates to Bug #280417, and the fix was rather simple:
Whatever updates occurred in the last day set all the devices to managed=false in /etc/NetworkManager/nm-system-settings.conf
Change that and run sudo killall nm-system-settings and it'll start working again.
I'm posting it here more as a note to self, than anything else.
And as I am in 'note-to-self' mode, it may be worth pointing out that, to configure a static IP address one has to edit:
iface lo inet loopback
iface eth0 inet static
# The following are the DNS servers for VirginMedia: