Hacham Ovadia (Yabia Omer Vol 6, OC 35) rules that it is prohibited to wind a watch that has stopped on Shabbat. He argues that this constitutes "Metaken Manah"-fixing a vessel. Although other Poskim are lenient, the Halacha is in accordance with Hacham Ovadia. The Menuhat Ahaba (Rabbi Moshe Halevi, Israel, 1961-2001, Vol. 3, P 255) permits instructing a non-Jew to wind the watch, since it is a disagreement between the Poskim whether a Jew may do it.
The Menuhat Ahaba also discusses whether a wind-up toy may be wound on Shabbat. He rules that it is different than winding a watch and may be wound on Shabbat. The reason is that the watch, when stopped, no longer serves its purpose as a vessel and is considered broken; therefore, winding it is a significant act of Tikun. On the other hand, the toy is still a viable play thing without being wound. The winding merely adds an additional dimension of activity.
It is prohibited to wind a watch that has stopped on Shabbat, but one may instruct a non-Jew to do so. It is permitted to wind a toy on Shabbat.