The problem, it turns out, was that the db_version field in the wp_options table and the wp_db_version variable in the file /wp-includes/version.php didn’t match. In my case (an upgrade to 3.5.2) the database showed “22441” while the PHP file showed “22442”. Changing the number in the database to “22442” solved the issue.
I’m not quite sure if this is a proper fix – I suspect the ‘Update Database’ script does more than just updating the db_version field. But… I can log in again and the dashboard tells me I’m using the latest version of WordPress.
For a more detailed explanation (with screen dumps) and a second possible solution, see www.clickonf5.org/2247/.