Not quite sure what bank account numbers have to do with a email address.
I think the OP refers to his/her email address being listed with his/her bank account details for authentication purposes etc.
We have several online accounts as well as utilities etc. When changing from BT five years ago we carefully went through all our affairs to remove our BT mail address and replace with our new email. As suggested on this forum we chose an independent provider, in our case Fastmail for about £23 pa. Our BT account still works although we left five years ago.