More shovelling to report. As above, my main site and network admin functions worked ok after running upgrade.php, but I later discovered my subsidiary sites remained in a state of complete torpidity.
I'm speculating, but I think the network update I tried last week must have timed out when everything was running dead slow. I tried it again from the network admin dashboard yesterday and it appeared to do nothing useful, so I logged out and manually applied upgrade.php to each subsidiary site in turn. This prompted the database update message for each site, and they seem to be accessible once more.
Have the perpetrators of this shambles been shipped to Guantanamo yet?