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. We ask Allaah to exalt his mention as well as that of his family and all his companions.
If it is confirmed that this man expressed his embracing of Islam but did not abide by its teachings, then he is not a Muslim in principle. Therefore, he is not suitable to be a husband to your sister, and it is not permissible for her to let him have sexual intercourse with her, otherwise she is a fornicator or adulterer. It is an obligation on all of you to advise her with soft words and good admonition while expressing eagerness to protect her interest. First, you have to seek the Help of Allaah then the help of whomever you think can influence her to whom she will listen, especially the scholars so that they can explain to her the religious ruling and the seriousness of this matter, which is a matter that no one can be tolerant about in any case. It might be that she will listen to you, especially since it seems that you have an important place in her life. Cutting relations with her or not depends on the benefit gained from doing so. If cutting the ties with her will push her to leave this man, it will be an obligation on you to cut relations with her. But if you are quite sure that cutting relations with her will lead her to become more sinful, then it will be more appropriate not to cut relations with her and continue to advise her.
Finally, we would like to draw your attention to the following three matters:
Firstly, the son will be attributed to them, if she and her guardian think that this man embraced Islam sincerely, as it is a doubtful marriage. The son will be attributed to his mother regarding religion so he is a Muslim and it is on obligation to nurture him and bring him up according to Islamic principles. The husband should not be allowed to influence the child.
Secondly, it is not permissible for a non-Muslim man to marry a Muslim woman, even if he is of the People of the Book practising his own religion.
Thirdly, it is not permissible for a Muslim to reside in a non-Muslim country except for a necessity.
Allaah knows best.