To answer my own question, when set up with an authenticator app, you have to use a back-up and restore to move to a new device. (If you've lost your phone you need to delete the number from your MS or gmail account otherwise you need to authenticate on the old device to restore onto a new device).
e.g. for MS Authenticator

Having spoken to several agents it seems there is no way to change your choice of 2FA, so if you chose SMS you have to keep using it, similarly Authenticator. BAEC agents can only reset passwords.