Since some Apple Watch does not have its own Internet connection, you must start the round with the phone around, or make sure the course is cached on Watch already.
Watch caches any courses it has ever seen. There are two ways to cache a course on watch:
Method A (from Watch):
1. Open Watch app, wait for it to search for the closest golf course. When the Watch looks like follows, you know the displayed course is cached.
Method B (from Phone first):
1. Open Phone app, wait for it to search for the closest golf course. Or click the location icon to search nearby golf courses, and select one of the ten closest golf courses.
2. Click "Begin Round"
3. Open Watch app, wait for golf course to sync over. Tap the screen to refresh if needed. When the Watch screen looks like follows, you know the displayed course is cached.
4. Optionally, after Watch displays the golf course, you can end the round on Phone. Or just leave it in the "Resume Game" state, and you will be ready to start the game any time even without Phone around.
Once a course is cached, you can start a new game on watch as long as the course is the closest one to your current location.