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.
If you have deleted the email account, when re- 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 which as you know are in this link.
https://www.bt.com/help/email/manage-email-account/manual-settings/what-are-the-settings-for-outgoin...
If the above does not help it appears that there could be a problem with the BT servers synchronising correctly. To try and over come this you should stop your devices from checking for email on the problem account. The easiest way of doing that is to turn the devices off or if you are not able to do that, change the email address by adding some characters to it so that the "real" account is not being polled.
Once you have done that, change your email password again and check that you can still access your email via webmail (on a browser).
If that works you should wait at least an hour and then add the new password to one of your devices and check it works. If it does then move onto the next device.
