The West Face scramble is the only known non-technical method to ascend the Watchman. It involves careful route finding to keep the difficulty at Class 3-4.
From the parking area at the end of the housing development off Canyon Springs Drive, follow a climbers trail toward the cliff base. Cross several small drainages and hike toward the southwest. There is one prominent section of a small lower vertical cliff band where it has collapsed- this allows for easy hiking onto the next level. The area of the upper ridgeline that you'll need to get to is described as: just south of Johnson Mountain's north summit. This is the point just north of Point 5875. Find the notch near here, via Class 3 ramps and gulleys. At the notch you'll notice a large sheltered pit that would make an excellent bivy spot.
Climb onto a small ledge and peer over and across the hidden drainage at The Watchman's West Face. You'll need to get an idea of the line you'll take from this point. Notice a prominent deep crack system across the way. Your line will start out either just to the right of this at the base (Class 3-4) or just to the left of this feature (Class 4). Scramble down from the notch into the drainage that separates Johnson Mountain from The Watchman. Near the upper part (north) of the drainage, cross over to the base of the west face of the Watchman. This section is the crux. You may find cairns near the two different slots. Above the crux slots, follow drainages up and start to contour left. If you encounter definitive Class 5 moves, your off route, and although they may go, you may still end up getting cliffed. As you contour left, you'll come to a prominent flat area where you may find a heavily cairned slope that leads out onto the edge of the west side of the upper cliff band. Soon you'll be at summit with tremendous views. Notice that the summit is not at the top of the prominent north buttress as seen from the valley.
A special thanks to cp0915 for his expertise on this scramble and the Zion area in general.