Hey Scritty,
I’ve never encountered this message, however checking Google’s FAQ, it looks like you can dismiss the warning. The page also explains why they may ask their users to change a password.
That being said, and in theory, all you should have needed to do was change your password, and log back in the old app. Once this was done, you should’ve received a message inviting you to migrate to the new app., which is an automatic process, and doesn’t require a code to do so.
As mentioned by Punyelf, please do read the pinned thread Authenticator Mobile App Migration Deadline Approaching for more information concerning the Authenticator migration.
If you still need help, you are always welcome to open a ticket.