Like it says in the title, I want to know if I have to manually update the mods or if it’s handled by the launcher that installed those mods.
No, mods aren’t updated automatically.
See https://github.com/remyroy/CDDA-Game-Launcher/issues/206 for some details.
The only sure way to get automatic mod update it to have those mods included in the base game. Even then, they might break from time to time if you frequently update to the latest experimental version.