In The Name of Allaah, The Most Merciful, The Bestower of Mercy.
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 Wudhu; rather she only washes her hands. That is because touching the private part without sexual desire does not make Wudhu obligatory. And it is well known that sexual desire does not 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 Wudhu is not obligated on her. [Source: Majmoo Fataawaa 11/203. Shaikh Uthaymeen (rahimahullaah). Slightly paraphrased]