Lockwise -- the new name for Firefox's built-in Password Manager -- should not be affected by a change to your IP address. Basically it just fills login forms for you.

Maybe it's a problem with Sync: if you compare between the two computers, do they have the same saved logins listed?

Oh, hold on a sec. If the other computer is not your own personal system but one that you share, perhaps you don't actually want your logins there at all. It might be better to use a different method for filling passwords that won't leave data on that computer.

You can install a portable (ESR) Firefox version if you want to use Firefox on a computer that isn't your own. The portable version comes with its own profile folder and doesn't interfere with currently installed Firefox versions.

• https://portableapps.com/apps/internet/firefox_portable