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 you were/are 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 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 15/30 minutes before starting your device and set up the email account with the new password.
Check that this device working before moving onto the next device if you have more than one.
Ensure that you are using the correct settings via this link and set the email accounts up manually rather than using the email wizard.
I would suggest that you now try changing the password using the sequence outlined in my first message.
I have come across this alot recently.
One trick that has worked for my clients with their Android and Apple phones, when you reset your password rather than using your @btinternet.com use @btopenworld.com instead.
However! If you have changed your password recently leave at least 3 days before trying this just incase it works. If not give this a try. It seems stupid but this has worked so far for 2 clients.
So you say "Forgot Id"
Type in your email with the BT Openworld ID on the back: "yourname"@btopenworld.com
Go through the recovery side, with either your phone or another email address. Phone is easier and if you have access to your BT account through a browser I would advise adding a number before you start this to speed up the process.
Then put in your new password.
You must then login to your email or ID through your browser first. Login in again with your @btopenworld.com name. If your sucessful this will login easy. Then log out and login with your @btinternet.com address using the new password. This should work.
Now try setting up your email on your phone. This should now work.
There seems to be a fault in updating the passwords on the server that looks after the Imap and Pop3 settings.
Let me know how you get on.