NO ROOT REQUIRED!
You can easily do this once you have installed and run the game.
Save Location PC:
C:UsersUsernameDocumentsMy GamesFallout Shelter
Save location Android:
They will be called Vault1, Vault2 or Vault3.
Rename the vault if you already have a vault on your other mobile device. Or overwrite your existing vaults.
I recommend using Dropbox to backup and transfer your saves when on the go.
I really doubt this works on…