I don't think you want to format the floppy, that will wipe everything from it. It should not make any difference what os is being used.
Are you sure it can't be read. It may be the type of file that isn't recognised. What files are they? The library pc will need the same program (probably) that they were created in. Also I'm surprised at any library allowing you to insert your own removable media.
It would also be a good idea to get all your work off floppies as soon as possible. They were hardly the most reliable of media, and anyway are now more or less obsolete. It's rare to find a new pc with one fitted now.