I think Authy has a version of their app for the Apple Watch. Not sure how it works in practice, but you should be able to use it for 2FA on websites as you'd expect. Of course, it probably still depends on your phone, like most Apple Watch stuff seems to.

Speaking of authentication and watches, I like the approach Apple uses, where it stays unlocked until it stops sensing a pulse, and then requires the passcode again. So it's convenient as long as you're wearing it, but it locks up when it's taken off.