How do I move the Microsoft Authenticator App inclusive existing account credentials to a new phone?
Prepare the old mobile phone
Open the Microsoft Authenticator app on your old phone. Go to the app's Settings (1)

Activate Backing up to iCloud (2) under the section Backup. This requires you to sign in with your private Microsoft account (Hotmail.com, Live.com, Outlook.com)
If Backup is already enabled, you can go directly to the section Prepare your new mobile phone.
Follow the on-screen instructions to back up your accounts. You will be asked to enter your Microsoft account password.
Prepare your new mobile phone
Important! Log in to the same iCloud account on your new phone that you used on your old phone. The backup from your old phone is stored in iCloud.
Install Microsoft Authenticator app from the App Store on your new phone.
Open the app and select Restore from backup (3) or Start recovery (4)
Important! Select either Restore from backup or Start recovery linkbefore you Logging in.
Sign in with the same private Microsoft account you used to back up your accounts on the old phone. Your backup will be reloaded.
Notes: If Restore from backup or Start recovery is not displayed when you open Authenticator, remove or log out of all accounts to restore your account credentials. See how.
Confirm successful transfer
After the recovery process, check that all your accounts have been transferred correctly to the new phone. You can now start using the Microsoft Authenticator app as normal.