You might want to check in outgoing server (SMTP) that the relevant outgoing accounts are set to actually use a password.

Thanks for your input.

Turns out that the server had blocked SMTP requests from our IP address but I was put off the scent by Thunderbird's pop-ups that suggested that this was a sign in problem. Once we were unblocked the transaction proceeded and a pop-up requesting the SMTP password appeared. This works but IMV it's really awkward that you cannot add a password without going through the transaction.

Once this was sorted I was able to discover the reason that we were blocked in the first place. One of Thunderbird's accounts was corrupted and no amount of fiddling with the SMTP parameters could untangle it. I resolved this eventually by deleting the account and starting afresh.