The issue BT had was a most likely a server problem. The best advice when it happens is to do nothing, and especially do not change passwords or delete the account from your phone as that can cause more problems than it solves.
The first thing to do when it happens is check to see if you can log on to your email account via a web browser. If you can there is obviously nothing wrong with your password or account hence the advice to do nothing.
If you also can not log into your account using webmail the advice again is wait and do nothing until you have at the very least checked through either calling BT or searching the Internet to see if there is a wider problem.
Where you have deleted the email account from your device or changed password you should stop your phone and any other devices from checking for email. The easiest way of doing that is to turn them off if you still have the account on your device or add a couple of characters to the email address so that it is not checking your account. If you have deleted the account and it is no longer on your device do not try and add it at this time.
Once you have done that, change your email password via webmail and check that you can access your email via webmail (on a browser).
If that works you should wait at least an hour to allow the servers to sync and then add the new password to your phone, remembering to remove any characters you may have added and again check it works. If it does work then move onto the next device.
When entering the settings, make sure that in the Outgoing server you do not use the "use same as incoming" for the email address and password. Enter them manually.
If the above does not resolve the problem, be aware when you change your BT password on an iPhone/iPad/Apple device it may only change for INCOMING mail.
To change the outgoing mail password you have to go into Settings; then scroll down to Passwords & Accounts or on older devices Mail, Contacts, Calendars. From there select the BT email account and select the account where it is listed under IMAP.
You will find the details for the account and can change the password for the Incoming server there, the outgoing mail server only shows SMTP and mail.btinternet.com You have to tap on that Primary Server in order to find where the Outgoing password is hidden so you can update it.
When setting up the email account on your device you are best to set it up manually, using "other" and ensure that you are using the correct settings as per this link.
https://www.bt.com/help/email/manage-email-account/manual-settings/what-are-the-settings-for-outgoin...
