I don't use BTInternet, but are you sure you have to "log on" to the mail server to send mail? If you are using a BT Internet connection, that should be enough.
Go to the Mail Settings panel in Thunderbird, scroll down to Outgoing Server (SMTP), and if it has "Use name and password" ticked, then untick it.
The smtp server should be the BT one (same as in your Outlook Express setup).
Try sending then.