That is not related to the fact that it is a local site; in other words, you can use the plugin and complete a login with success in a local installation of Wordpress that is not available on the public network. There’s some limitations in configuration options of the plugin if it’s not a public instance, but that would be irrelevant in terms of completing a login after configuration is completed.
As you mentioned the most likely cause would be date/time settings leading to problems. The Auth0 server machines that issued the tokens have a service specifically configured to ensure the proper date/time at all times (Network Time Protocol - Wikipedia) so this would point to an issue in the client settings. You should review the date/time configuration again because from what you mentioned there’s an issue due to the offset not being correctly shown; this would likely imply the instance is not configured correctly in terms of timezone.