Everything should have carried over. If not, you may have cleared your browser storage before this update launched by accident. I can see people are uploading worlds so it's not completely broken, but let me know if your world specifically is erroring out when trying to publish.
I can't speak to the other operating systems, but If you're using Windows then I believe in previous versions worlds were saved in the AppData/Roaming/Formamorph folder for the user account you're running the app from. If you're using itch sandbox, then that would be the 'itch-player' user, otherwise it would be the profile you're signed in with when playing. (The AppData folder is also hidden by default, so you'll have to enable viewing hidden items to see it.)
If the current update doesn't clear anything from that folder, then downloading an older version should work since the files should still be there.