Thanks, but no joy.
1 - Switching the theme by renaming the current themes folder just results in a blank page, even after trying to force a login.
2 - Resetting the plugins folder by fiddling the database has no effect. The same No Update Is Necessary page still appears when I try to go to the dashboard.
3 - Rerunning the upgrade using wp-admin/upgrade.php has the same effect as your second suggestion.
It appears to be running 3.1 because I can see the new (ugly) admin bar atop the front page. ...but it's pretty useless 'cause there's no way to add content without being able to get to the dashboard. Choosing anything from the new admin bar just brings up the No Update Necessary page again.