Praise be to Allah.
It is permissible for a women to breastfeed her child before doing ghusl, whether it is ghusl after nifaas, menstruation or sexual relations. There is no evidence to indicate that she must do ghusl in these cases in order to breastfeed her child. Rather ghusl is obligatory in cases of impurity in order to pray or to do any act of worship for which tahaarah (purity) is required. The woman who is bleeding following childbirth or menstruating must do ghusl when the nifaas or period ends in order to do those acts of worship and also so that it may become permissible for her husband to have intercourse with her. It is not haraam for the woman who is menstruating or bleeding following childbirth to touch anything during her period or nifaas, except for the Mushaf (copy of the Qur’aan in Arabic only). And Allaah knows best.