How to prevent the Apple Watch from going to sleep?

As of my knowledge and other searches on this topic, at the moment is no api available to enable or even disable sleep mode for Apple Watch programmatically.

No-Sleep API

Currently, there is no API to disable Apple Watch from going to sleep.

You may wait until watchOS 3.

Changing Sleep Time By User

Any user can go to this path:

Settings -> General -> Wake Screen

And set the time to a long time.

Unfortunately, only the user can change this and there is no way to do it programmatically.

NOTE: If you want the screen to resume your activity after waking up, you should ask user to do so here:

Settings -> General -> Wake Screen -> Resume Last Activity

Note that according to Owen, keeping the watch awake would kill the battery quickly.

Conclusion

1- No API is set.

2- User can change sleep time.

3- User can select to resume last activity when the Apple Watch wakes up.

Because of the power requirements, keeping the watch awake would kill the battery quickly. However, you may be concerned that when the watch does go to sleep, it doesn't re-wake to your app. This can be remedied at the user level, on your iOS device using the Watch App, using General->Wake Screen-> and checking "Resume Last Activity"