There have been problems with the BT IMAP/POP servers stopping email clients from accessing the users email account for years. The advice from forum users is to do nothing such as changing passwords or deleting and resetting up accounts as this invariably causes more problems than it solves.
If all your devices stopped accessing your email at the same time and you were still able to access your email via webmail on a browser it is obviously not a problem with your email accounts or with your password, hence the advice not to do anything. The problem usually resolves its self in a day or two.
Because you have changed your password I would suggest that you do the following:
Switch off all your devices or at the least stop them from polling your email account for new email. You can do this by either deleting the email account from the device or just change the email address in your email account settings.
Change your webmail email password yet again.
Wait at least 30 minutes before starting one of your devices and set up the email account with the new password.
Check that this device s working before moving onto the next device.
Ensure that you are using the correct settings via this link and set the email accounts up manually rather than using the email wizard.
https://bt.custhelp.com/app/answers/detail/a_id/44917/kw/bt%20mail/c/6591
Looks like this has reared it's head again, online chat today they verified it's a server problem when checking out my email address. Supposed to get a call back within 12 hours...
This problem has been there for years and has never gone away. Just some customers seem to have more problems than others
there are numerous posts about email problems and the advice is always the same - do nothing and wait a few days and try again. Access is normally is always available using web mail