If you access Tools >options > Security > Passwords

and the password for both the Mailbox and smtp for that account is definately not there, then restart Thunderbird to see if Thunderbird prompts for password.

Done that. No change. Problem remains unsolved.

Can you double check in your webmail account accessed via a vrowser, those accounts are selected for either 'Pop or IMAP forwarding' and that the type of account you selected is the type of account created in Thunderbird.

Then post this info...edit all font and print info but do not edit anything else. https://support.mozilla.org/en-US/kb/ask-tb#w_how-to-ask-your-question

Thank you very much for your time and trouble, Toad-Hall. Problem solved.

For your information, I accessed Tools >options > Security > Passwords and deleted ALL the passwords (both ingoing and outgoing) associated with the mail accounts for which I had changed the password directly via cPanel. (Because the 3 accounts are on the same domain, Thunderbird was unable to download or write to all 3 of them, as I had changed the passwords.)

I then closed Thunderbird and restarted my COMPUTER (merely closing and restarting Thunderbird was insufficient for Thunderbird to bring up the password prompt window).

Doing both those things solved the problem. I opened Thunderbird, selected each account in turn and clicked "Get mail". Pop-up window asking for the password came up, I input and was able to write and download messages. I repeated for the other 2 accounts, and now all works fine.

Thanks so much for your help.

Just curious, but does your anti virus program have some sort of password "vault"