Installing isn't working
A few things worth checking before assuming something's actually broken.
About 3 minutes
A few common causes, roughly in order of how often they turn out to be the answer:
Check your watch face picker before anything else. Occasionally a face finishes installing on your watch, but the confirmation message back to your phone doesn't arrive in time, and the phone shows a failure that didn't actually happen. Your watch face picker is the source of truth. If the face is there, it worked.
No internet connection. Buying a face needs a live connection at the moment you tap Get, on whichever device you're using. There's no offline purchase queued up to retry later. Check your connection and try again.
Installing from your phone, and nothing happens on your watch. Make sure your watch is paired, nearby, and connected over Bluetooth: the phone sends the install to your watch over that connection, and if your watch is out of range or asleep, it won't arrive. Once your watch is back in range, try again from your phone, or install straight from the watch app instead.
The face appeared, but didn't become your active face. If Face Hub hasn't been given permission to activate faces automatically on your watch, you'll see a one-time prompt instead: press and hold your current watch face, then pick the new one from the picker. That's normal, and it's only a one-time thing per watch.
Still stuck? Try installing that same face directly from the watch app. It doesn't depend on the phone at all, so it's a good way to tell whether the problem is the phone-to-watch handoff or something else.