Fataawaa Regarding Women: Is a woman required to perform Wudhoo after giving her son or daughter a wash? Shaikh Uthaymeen (rahimahullaah)
If a woman gives her little son or daughter a wash and touches the private part of (the child), it is not obligated on her to perform Wudhoo; rather she only washes her hands. That is because touching the private part without sexual desire does not make Wudhoo obligatory. And it is well known that sexual desire does come into the thoughts of a woman who gives her children a wash; so after she gives a little girl or boy a wash, she only washes off the impurity on her hands, but Wudhoo is not obligated on her.
[Source: Majmoo Fataawaa 11/203. Slightly paraphrased]
Abu Mu’aawiyah (Abdullaah Al-Gambi)