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 the situation is as you mentioned, then that woman has become the wife of that man because the marriage of non-Muslims is held valid. Ibn Qudaamah said:
“The marriages of the disbelievers are valid and are approved of if they become Muslim or if they refer to us for judgment if the woman is one whom it is permissible to marry at the present moment. The details and manner of the marriage contract are not to be examined and the marriage is not to be examined to see whether it conforms to the conditions for marriage among the Muslims, such as the presence of the Wali (woman’s guardian) and witnesses, the format of the proposal and acceptance, and so on. There is no difference of opinion concerning this among the Muslims. Ibn ‘Abd al-Barr said, 'The scholars unanimously agreed that if two spouses become Muslim together at the same time, they remain married according to their original marriage so long as they are not unlawful to each other because of blood relations or breastfeeding. People became Muslim at the time of the Prophet and their wives also became Muslim, and their marriages were approved of; the Prophet did not ask them about the conditions of their marriage or how it was done. This is something that is known through Tawaatur (i.e. being reported by a group of narrators from a group of narrators, generation after generation and so forth, and they are all trustworthy people and it is absolutely impossible for them to agree on a lie), and it is something that is well-known to all Muslims, so it is certain.”
The fact that the marriage was conducted in order to get residence does not affect its validity.
As long as that woman is still in a marriage with her husband, it is not permissible for anyone to marry her unless her husband separates from her by invalidating the marriage or by divorce; because marriage [the fact that a woman is already married] is an Islamic impediment [for marrying her]; Allaah says in the context of mentioning the women whom we are forbidden to marry (what means): {And [also prohibited to you are all] married women.} [Quran 4:24]
It should also be pointed out that in regard to a Muslim man marrying a woman from the People of the Book, it is a condition that she is chaste. Also, it is better for a Muslim man to marry a Muslim woman because Islamically there are disadvantages in marrying a woman from the People of the Book.
The Commentary of Ad-Dusooqi, from the Maaliki School of jurisprudence, reads, “Imaam Maalik disliked this in the Muslim land because she feeds on wine and pork, and she feeds her child with them, while he kisses her and sleeps with her, and he is not entitled to forbid her from being fed with that, even if he is harmed by its smell, and he cannot prevent her from going to the church, and she may die while she is pregnant and she will be buried in the graveyard of non-Muslims, which is a pit of Hellfire.”
For more benefit, please refer to fataawa 84724 and 88451.
Allaah knows best.