Thunderbird may not even get to the point to prompt you for your credentials, e.g. when the server cannot be reached. Check your account settings and network connectivity.

Somehow it seems that my other account remains open. The password process seems to have made itself redundant.

How can I reinstate it?

I don't think I understand what you're trying to achieve.

I have 2 email accounts with separate email addresses. When I open Thunderbird, usually I have to login to both of them.

Now, I only have to login to one of them, and I can operate both accounts.

Even if I don't login, the second one, whose login box has disappeared remains open; I can read emails, recieve and send.

How can I reinstate apssword protection?

Delete the password Thunderbird has remembered. At the top right of the Thunderbird window, click the menu button > Options > Security > Passwords > Saved Passwords