I'm running Kubuntu 14.04 (which means KDE 4.13.3), and the latest Firefox release (FF46) hoses the file selector (see https://bugzilla.mozilla.org/show_bug.cgi?id=1268395). This would be due to a KDE bug that has been fixed (but obviously after the 14.04 code freeze): https://bugs.kde.org/show_bug.cgi?id=330596.

What is the most convenient way to upgrade my system to fix this? Or is the fix going to trickle its way down to Ubuntu 14.04?

You may want to add kubuntu backports repository and check if they pushed this fix. If not, you can ask on kubuntu developers mailing list but I doubt if they be willing to do that because they are rather busy with Kubuntu 16.04 and Plasma5.
If I were you I would delete .mozilla (just rename or do backup first) folder in your /home. This will reset Firefox settings to default. I am not sure if it can change anything but worth checking. I also use Kubuntu 14.04 and I don't experience this problem. But I use adwaita both for gtk2 and gtk3 apps.
This might interest you
https://www.kubuntuforums.net/showthrea ... -UI-issues