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 is His slave and Messenger.
If what you mentioned about that woman is true, then she has indeed committed a number of evil acts, starting with running away from her husband, establishing an affectionate relationship with a non-Muslim man, and the most serious of which is apostating from Islam. By apostating, her marriage contract becomes void; for more benefit, please refer to Fatwa 99158. Therefore, she should be reminded of the seriousness of what she did, and she should be advised to repent and re-embrace Islam.
In principle, an apostate should be deserted and one should not socialize with him/her as long as he/she is still an apostate. However, if there is a benefit in socializing with him/her, like hoping that he/she would re-embrace Islam, then in this case, it is permissible to socialize with him/her for this purpose. Hence, we advise you to try to bring this woman back to Islam by being kind and tolerant with her hoping that Allaah will make you a reason for guiding her. For more benefit, please refer to Fataawa 94482 and 87672.
As regards the child that she gave birth to, then in principle it is attributed to her husband as he is the legal husband if he is born at a time when it is possible that he would be from him, in this case, he would not be able to deny him [the child] except with Li’aan (i.e. when a husband and his wife invoke the Curse of Allaah on the liar amongst them in case he accuses her of having committed adultery). Therefore, the child is a Muslim as he follows his father in religion, because the child follows the parent who has the best religion, as discussed in Fatwa 81957.
Finally, if this woman continues to be an apostate, then it is not permissible for her husband to leave the child with her so that she would not spoil his/her religion.
Allaah Knows best.