OK, I seem to have this working as I'd like. Thought I'd post this in case it helps anyone.
I currently have a USB ADSL modem so it's broadband but not strictly an 'always-on' setup. However, I am running the AutoRasDial service so that makes it an 'always-on' setup, i.e. my connection is made at boot up. So my internet connetion is active even when no one is logged on.
I have select 'my computer is always connected' in Avast!.
The behavior is that the computer boots. Network connects and Avast! services start. If I logon 10 minutes later I immediately get a msg that Avast! has updated. When I check the log I find that it did so within seconds of booting. Very reassuring. So with this setup I can have confidence that even if the PC is left logged off for hours Avast! will still be able to keep itself updated. (A bonus is that my BOINC service is using the connection with no user logged on too.)