Your Apple TV doesn't actually use the login you used - it uses your broadband line to identify you. Your iPhone needs a valid account though.
I can see that you used to have a BT account, but you've moved to EE Broadband. Your EE account isn't currently associated with an EE ID - it's ready waiting to be claimed.
I can also see you already have an EE ID associated with your mobile phone contract. Simply visit ee.co.uk, select the "Log in" link in the top right corner, and then log in with your EE ID (or you're welcome to create a new one if you'd prefer). Once logged in, scroll to the very bottom of the page - you'll see a button to "Link a product". Select that, and add your new EE Broadband account.
Once linked, you'll then be able to log into the EE TV app on your iPhone - just make sure you choose the EE Broadband option when offered.
