I'm not sure exactly what you're saying, but it sounds like all you need to do is change the email address in the accounts you have instead of adding a new account. That would include putting in the info for smtp server.

Thanks for your reply. The e-mail address is already in the accounts (it is unchanged). What has changed is the server where the e-mail domain resides. Somehow I need TB to take note of that. I have also tried deleting the password. It duly asks for a new one, so I give it the password for the new server but doesn't connect. If I put the old password in TB tells me there are no messages, so it seems it is still connecting to the old server (though I wonder why, since the nameservers have been changed).

Later: Next time I tried TB threw up a notice about not liking the SSL certificate (which is on the new server), so I allowed an exception and it is now connecting. I can only assume it just took the nameserver sources that TB uses longer to get the message (of the ns change) than others. Problem solved.

I really don't think you should need to do anything as you have migrated the server at the DNS level. My guess is the mail was routed earlier as changes to DNS propagate and the backbone connected providers like google, yahoo amazon AWS, cloudflare et. al. getting changes first and their subordinate DNS servers getting changes from them means it can take days for such a change to reach the more inaccessible corners of the internet.

I think that's right. Eventually all I had to do was change the password. I could even have avoided that by setting the same password at the new server, but Plesk (the hosting control panel) insisted on more symbols in the password than I'd had before.