If Willow Satyr leaves the battlefield, you no longer control it, so the duration of its control-change effect ends.

Once the ability resolves, it doesn’t care whether the targeted permanent remains a legal target for the ability, only that it’s on the battlefield. The effect will continue to apply to it even if it gains shroud, stops being a creature, stops being legendary, or would no longer be a legal target for any other reason.

If Willow Satyr stops being tapped before its ability resolves — even if it becomes tapped again right away — you won’t gain control of the targeted creature at all. The same is true if you lose control of Willow Satyr before the ability resolves.