It is permissible for a non-Muslim woman to breastfeed a Muslim child, because the basic principle is that it is permitted, and there is no evidence to the contrary. Likewise, a Muslim woman is allowed to breastfeed a non-Muslim child.
If breastfeeding takes place, then the Islamic ruling on each of them is not altered because of the breastfeeding. Whoever was a Muslim before breastfeeding remains a Muslim afterwards and whoever was a non-Muslim before breastfeeding likewise remains a non-Muslim.
In his response to your question, Sheikh Atiyyah Saqr, former head of Al-Azhar Fatwa Committee, states:
The human being before puberty is a follower of the nobler religion of his parents; so if, for example, his father is Muslim while his mother is Jewish or Christian, he is considered Muslim. And if his father is Magian while his mother is Jewish or Christian, he is Jewish or Christian according to the religion of his mother whose religion, in such case, is nobler than his father's.
Upon puberty, a boy or a girl becomes legally competent, and from then on is regarded a Muslim, a Christian, a Jew, and so on, according to what he/she was before puberty. If he/she abandons the religion, he/she is then an apostate. All these judgments pertain to blood paternity or maternity. While the paternity or maternity because of breastfeeding have certain judgments in marriage and its related issues, these judgments are not related to the breast-fed child's religion. Thus, it is permissible for a Muslim child to be breast-fed by a non-Muslim woman because there is no evidence of prohibition. This is especially so if there is no one else to breast-feed him except this woman. If that child is breast-fed by her, he is still a Muslim; likewise, if a Muslim woman breast-feeds a Jewish or a Christian child, this child remains Jewish or Christian.
Therefore, the Muslim man who was breast-fed by a Christian or a Jewish woman is prohibited to marry her because she is considered his mother because of breastfeeding. He is allowed to touch her, sit with her alone, and look at her while she is without a veil, and so on, just as with his real mother in suchlike matters. As for his religion, it is not affected by this breastfeeding.