If you have internet connectivity and you can receive email from your other address(es) then it is likely down to settings.
Have 1&1 changed the name of their incoming/outgoing mail servers ?
When you say mail out is unaffected as all email accounts are operating - are you sure your outgoing email isn't just going out via BT ? Have you tried doing a send only on 1&1 ? It may be that your 1&1 account details need re-setting up.
You could try pinging their mail servers that you have setup in the 1&1 account details as a first test.