The "wilderness protocol" or maybe some personal variation of it, would require some fairly precise timekeeping, or you would totally miss the communications window or else waste batteries by monitoring or transmitting when the other party isn't listening.

As someone else mentioned, knowing how long until it gets dark (OK, maybe you don't know sunset on the first day, but from there on out, you would know) will help you plan your actions. If it's three hours till sunset and your next travel leg is at least four hours, then maybe it's best to stay put. A similar situation is if there is some dawn-to-dusk curfew in place and you really don't want to be caught outside when the curfew is in effect. Obviously, you can tell that the sun is going down by just looking at it, but a watch will give you a better indicator of the amount of daylight left when the sun is still relatively high in the sky. Maybe some of us can eyeball the time this way, but I've never developed the skill. If you need to set up a rotating watch, e.g. taking turns watching a nearby river through the night during flood conditions, then having a watch keeps things equitable.

When purifying water, obviously a watch will help you purify the water long enough, whether through boiling or chemical disinfection. But it would also help you from needlessly wasting fuel or waiting too long for chemical disinfection when you're really in need of water as soon as possible. And if you need to repeat the treatment, e.g. using chorine bleach drops, then you could waste a lot of time going through the process a couple times without having a reliable indicator of time.

Weather warnings are usually in effect for set amounts of time. If you really want to go out on the road but you know that there is a flash flood warning for three hours, you can stay put for the shortest time without unnecessarily putting yourself in danger. The warnings are always broadcast when they are issued, but there's often no broadcast when one expires, so you have to keep track of them yourself. Or maybe you know about the alert when it's issued, but don't have a radio/TV handy when it expires.

Anyway, just some random time-related activities.