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, as you have now found out.
Because your devices stopped accessing your email and you were still able to access your email via webmail it is obviously not a problem with your email account 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 deleted and reset up your email on your device and changed your password I would suggest that you do the following:
Switch off all your devices that use that email account 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 is 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
