In a recent thread, this problem was linked to Firefox's Tracking Protection feature (https://support.mozilla.org/questions/1492036). To test whether that is the issue for you:

When you are on the main page for the site, turn off Tracking Protection by clicking the shield icon at the left end of the address bar, next to the lock icon. On the panel that drops down, click the slider switch to disable protection. A slash should appear through the shield.

If the login page is on a different server, the shield will not have the slash. Repeat the above process for the server on which you sign in.

If the account management page is on yet another server, the shield will not have the slash. Repeat the above process for the third server.

At that point, hopefully Firefox will allow your credentials to be shared across the bank's servers so you can stay signed in everywhere. Until you sign out to secure your account.

Does that work for you?

The building society says it is is due to a recent Firefox update being insecure. Can Firefox fix this?!

Well is news to us if that was indeed the case. While this does not appear to be the case with https://www.coventrybuildingsociety.co.uk/ the Firefox web browser will tell you if the server certificate was expired while other web browsers based on Chrome/Chromium may not tell you for say a week.

And since there's no link to report that as a Bugzilla for error issue there nothing Mozilla can do about that.