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Go to Administer/User and Permissions/Permissions (Access Control). The click on the top option Drupal/Wordpress/Joomla Access Control. Make sure anonymous users have "CiviCRM: access all custom data " checked
You can refresh the page. Sometimes the upgrade will just timeout on one task, but a refresh will restart the timer. Also have you tried running the upgrade using CV on the commandline?
How frequent are your Drupal Cron and CiviCRM Cron? AND is Clean Up Temporary Data turned on? Typically I see this error when the form is loaded but a session cache is cleared between the load and the submit. Temporary data only flushes data that is already old, but if the timestamps are not the same timezone as the site it can flush all current data. Check the logs for a form load/ cron run/ then submit failure pattern.