Al-jawab billahi at-taufeeq (the answer with Allah's guidance)
No, the Wudhu will not be nullified if you do not wear a scarf whilst performing Wudhu or outside Salah. However, not wearing the scarf or covering the Satr (A female's Satr is her entire body except her face and hands to the wrist) in Salah, will invalidate the Salah.
Also, it must be remembered that to wear the Hijab/ Niqab outdoors is compulsory and indoors if there are Ghair Mahram men (those whom you can marry) present.
