All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad, sallallaahu ‘alayhi wa sallam, is His Slave and Messenger.
An illegitimate child born out of wedlock bears the name of the mother only and has no legitimate relationship with the biological father who committed adultery with the mother according to the Sharee‘ah. Please refer to Fatwa 84788.
The child in this case assumes the religion of his mother so he is regarded as a Muslim, given that the mother was originally a Muslim, whether she was a pious or a non-practicing Muslim.
This mother is not considered a Christian but rather an apostate. In his commentary on the statement of Khaleel: “The apostate’s child remains a Muslim”, Al-Kharashi wrote: “This means that if the apostate is executed while persistently adhering to apostasy, his young child is still considered a Muslim and is not to be considered an apostate like his father because the child takes his father’s religion only if he follows a divine religion acknowledged by Islam (Christianity or Judaism). The child in this case is considered a Muslim whether he is young or has reached puberty and whether he was born before or after the father’s apostasy.”
Hence, the mother's conversion to Christianity is irrelevant in this regard, whether she became an observing Christian or not.
When the child grows up and gets married, his children take the same ruling as him, i.e. they are considered Muslims like their father.
Allaah Knows best.