After lookng at all the current open issues and since this is not implemented I am raising this issue for the firebase_auth plugin: https://github.com/flutter/plugins/tree/master/packages/firebase_auth

The functionality exists as part of firebase:
https://firebase.google.com/docs/auth/web/manage-users#set_a_users_email_address

Much like changing the users password (#16775) this should also be possible.

Raised against the plugin: firebase_auth up to 0.5.8