I find it odd that the author hasn’t yet responded in this thread yet is active in others.
I had the same problem using an older version of the plugin and the feed stopped updating on June 11th. I updated the plugin and went through the process of creating a twitter application, generating the various keys and authorising the account.
Make sure when creating the account to enter a callback url – your site homepage will do or you will find you can’t authorise the account and you will be stuck on the settings page scratching your head! I couldn’t see how to edit the app I’d made in twitter to amend the callback url so deleted it and started over.
If you are updating an older version like me then you will need to go to the widget and enter the username and list details. Ignore the message that the account needs to be authorised as once you save the widget this will disappear. Hope this helps somebody – mine is currently showing the recent tweets. Of course I don’t yet know if it will update long term but hope it’s now fixed!