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.
What that man and woman did, i.e. having sexual relations as a husband and wife is a grave major sin as this matter of taking secret lovers is a matter of the pre-Islamic era of ignorance which Islam had abrogated and forbade. Allaah says [addressing men] (what means): {…nor those who take [secret] lovers.} [Quran 4:25] Allaah also says [addressing women] (what means): {…or taking [secret] lovers.} [Quran 5:5] Therefore, they are both required to repent to Allaah sincerely. For information on the conditions of repentance, please refer to Fatwa 86527.
If that woman has become Muslim while her husband remained a non-Muslim, then her marriage to him has become void, and it is not permissible for her to have sexual intercourse with him or allow him to have sexual intercourse with her; rather, she is obligated to separate from him. When her waiting period from him expires, it becomes permissible for her to marry someone else. For more benefit, kindly refer to Fatwa 107894.
However, it is not permissible to marry a woman who had committed Zina (illegal sexual intercourse) except after she sincerely repents and after making sure that she is not pregnant from Zina as we have clarified in Fatwa 82718.
What you have mentioned regarding the non-existence of a law about divorce or separation, strengthens the fact that one should not marry that woman in order to avoid finding oneself in an embarrassing situation by being legally questioned by the authorities if that Christian man claimed that she is still his wife.
However, if she became pregnant after her marriage with her non-Muslim husband had become void and her waiting period had expired, then the child is an illegitimate child that should be attributed to its mother as clarified in Fatwa 97254. But if she had become pregnant before her waiting period expired, then the child is attributed to her Christian husband as the Prophet said: “The child is for the owner of the bed.” [Al-Bukhari and Muslim]
In any case, this child is ruled to be a Muslim as he follows his mother in regard to religion because the child follows the better religion of either of his parents. For more benefit, please refer to Fatwa 81957.
Finally, the best solution for this marriage to take place in the best manner is that that woman endeavors to prove that she has become unmarried and she should try all possible means to do so. Besides, if she has no Muslim guardian, it is the Muslim ruler who would marry her off or whoever could act on his behalf, like Islamic Centers. For more benefit, please refer to Fatwa 88488.
Allaah Knows best.