Depends on the app.
My Outlook app (which, unfortunately, I'm required to use for work) does not require iCloud as it syncs directly with our Outlook server.
Some Apple apps like Notes and Reminders will depend on where your notes and reminders are stored -- if they're iCloud-based and you're away from your phone (like on cellular), then it'll pull them off iCloud. If they're local phone-only, then it doesn't run through iCloud. If they're, say, Google-based, then I'm not sure what the watch does (if it pulls them from the phone or if the watch communicates directly with Google's servers; I'm guessing the former).
But the watch will work even if you're not using any iCloud services. You may be more limited in what it can do when you're away from your phone, but otherwise there shouldn't be much if any loss of functionality.