Mail unable to verify user name or password

I've had the same Mac for years and the same BT E-mail account here in the UK. This morning all accounts were offline, and when I went to bring them online, I got the message 'unable to verify user name or password'. The password box is empty. I've entered it but repeatedly get the same message. I went online to BT.com, entered the password and accessed my mail online, but Mail will not access my account or download the mail to my Mac. I've spent half an hour with a BT representative through chat who claims Mail is a 3rd party application and since I can access mail online it's not their fault.

BT wanted me to delete Mail and reinstall, but I can't find out how. Even if I do, I don't want to lose all of my e-mails and data such as addresses. It's an IMAP account at mail.btinternet.com.

I still can't download mail or access the account after almost eight hours today. Any thoughts?