The Ben Ish Hai (Rav Yosef Haim of Baghdad, 1833-1909) writes that mothers should wash the hands of their young children in the morning – even children who have yet to reach the age of Hinuch (training in Misva observance). Firstly, he explains, young children touch food, and if their hands are not washed in the morning, the foods they touch will become contaminated by the Tum'a (impurity) that descends upon the hands overnight. Secondly, washing the hands of young children in the morning has a significant spiritual impact, as it helps to lead them toward the path of purity and holiness.
By contrast, Rav Shneur Zalman of Liadi (first Rebbe of Lubavitch, 1745-1812) writes that the hands of young children do not become impure overnight. Before a youngster becomes Bar-Misva, Rav Shneur Zalman explains, he does not have a pure Neshama (soul) that leaves during the night which would cause the forces of impurity to descend upon him. Therefore, according to Rav Shneur Zalman, there is no requirement to wash a young child's hands in the morning. Similarly, the Sitz Eliezer (Rav Eliezer Waldenberg, Jerusalem, 1915-2006) writes that the prevalent custom is to be lenient in this regard, and not to ensure to wash the hands of youngsters below the age of Hinuch.
However, Hacham David Yosef, in Halacha Berura, writes that nowadays, when washing hands is a very simple matter, there is no valid reason to be lenient, and parents should ensure to wash the hands of even their young children.
If a mother is a Nidda, it is preferable for her not to wash her young children's hands in the morning. Rav Yisrael Bitan writes that at first glance, we might think to allow her to wash the children's hands, since this washing is not required according to the strict Halacha, and so we do not need to treat it stringently. In truth, however, once we have established that it is proper to satisfy the opinion that impurity indeed descends upon the hands of young children, the washing should be performed in a manner that removes the Tum'a without any uncertainty. Rav Haim Palachi (Turkey, 1788-1868) writes that water poured by a person who is Tameh cannot remove the Tum'a from the hands, and so when a mother is a Nidda, the children should pour the water over their hands themselves if possible, and if not, then the father should pour the water. The water should not be poured by an unmarried girl who has already reached adolescence, as she, too, is a Nidda. Rav Haim Palachi writes also that the washing should be done by a Jew, and not by a gentile, such as a non-Jewish housekeeper or nanny.
The Yalkut Yosef writes that a woman who is a Nidda is allowed to pour water over her children's hands, citing this ruling in the name of the Be'er Moshe (Rav Moshe Stern of Debrecen, 1914-1997). The Be'er Moshe reached this conclusion based on the Halacha permitting a Nidda to pour water on others' hands for Netilat Yadayim before eating bread, assuming that this should apply as well to the morning Netilat Yadayim. Rav Yisrael Bitan noted, however, that the morning hand-washing involves the removal of impurity, and thus differs from the washing before a meal. As mentioned, several sources indicate that a person with impurity is incapable of removing the impurity from another person's hands in the morning. Rav Bitan thus concluded that although those who wish to be lenient have a basis on which to rely, it is preferable for a woman who is a Nidda not to wash her children's hands for them in the morning.
More generally, the Kab Ha'yashar (Rav Tzvi Hersh Kaidanover, d. 1712) writes that if a person wishes to have somebody else pour the water over his hands for Netilat Yadayim, he should ensure to have this done by a G-d-fearing, religiously-devout individual. The Zohar teaches that each morning, Eliyahu Ha'nabi goes to Me'arat Ha'machpela to wake up the Abot (patriarchs), and he washes their hands so they can pray. The fact that Eliyahu was chosen for this role, the Kab Ha'yashar, writes, shows that washing should be done by a righteous person.
Rav Bitan emphasized that washing the hands of young children in the proper way is important for their religious growth. The Shulhan Aruch (Orah Haim 4:18) writes that failing to observe the requirement of Netilat Yadayim can cause a Torah scholar to forget the material he has learned. Rav Haim Palachi explains that Torah knowledge can be acquired only if one is in a state of holiness, and therefore, one who compromises his holiness by failing to wash his hands as required cannot retain his Torah knowledge. Indeed, the letters that precede the letters that spell "Torah" are Shin, Heh, Kof and Dalet – which spell the word "Kedusha" ("sanctity"). This shows that holiness is a precondition for Torah. If parents want their children to absorb Torah and grow in Torah learning and observance, they must ensure to maintain the children's state of purity, and this necessitates washing their hands each morning, even when they are very young.
The Hazon Ish (Rav Avraham Yeshaya Karelitz, 1878-1953) is said to have taught that one who seeks to retain his Torah knowledge must ensure to always speak truthfully, to recite the traditional "Yehi Rason" prayer before learning, and to meticulously observe the laws of Netilat Yadayim. In fact, it is told that when the Hazon Ish was a child, his family ensured that all the children had their hands washed in the morning already from a very young age.
Rav Haim Palachi noted that proper observance of the Netilat Yadayim requirement also ensures that one will always have a sufficient supply of water, even in times of drought, and will succeed in maintaining the purity of his thoughts.
Summary: From a strict Halachic standpoint, it is questionable whether young children below the age of Hinuch (training in Misvot) must have their hands washed in the morning. Nevertheless, this should be done, as this contributes to their spiritual growth. A woman who is a Nidda – including the mother – should not wash the children's hands, and neither should a gentile, such as a non-Jewish nanny.