Praise be to Allah.
It is permissible for a Muslim woman to breastfeed a Christian child and it is permissible for a Christian woman to breastfeed a Muslim child, because the basic principle is that that is permitted, and there is no evidence to the contrary. Rather that is a form of kind treatment and Allaah has decreed kind treatment in all things. It is proven that the Prophet (peace and blessings of Allaah be upon him) said: “In every living being there is reward.”
Secondly: if breastfeeding takes place, then the Islamic ruling on each of them is not altered by this breastfeeding. Whoever was a Muslim before breastfeeding remains a Muslim afterwards and whoever was a Christian remains a Christian.
May Allaah send blessings and peace upon our Prophet Muhammad and his family and companions.