You should be fine. Minecraft has a built-in DataFixerUpper that will upgrade any old assets to the newest version, though it’s suggested to take backups before updating.
I got it to update now. My problem was that when I updated to 1.21 it would start the world from scratch. Not sure what the exact steps are to do it correctly but when I shut the server down and started it back it up, it went from 1.20.6 to 1.21.