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 you married this woman after she pronounced the two testimonies of faith or when she was from the people of the book, and the marriage fulfilled the conditions of a correct marriage, like the presence of the guardian and the two witnesses and the like, then she is your wife and your children from her are traced back to you. So you should be kind to her by teaching her Islam and its bounties, especially by setting an example by your conduct with her. Besides, you should incite her to observe the religious obligations. It is not a condition for her becoming a Muslim to announce the two testimonies of faith in the mosque. If she utters these two testimonies of faith and does not commit anything that contradicts them, then that is enough to judge that she is a Muslim. You should take her to another environment where she can practice her religion without any harm.
But if you married her while she was a non-Muslim and does not follow or believe in any religion, or that one of the conditions of marriage was not met, like having married her without the consent or presence of her guardian, then she is not your wife and you should divorce her. As regards the children, if you thought that your marriage with her is correct, then they can be traced to you. If she embraces Islam, then there is no harm for you to marry her after making sure that she is not pregnant [one menstruation], but if you fear that she might not be sincere in becoming a Muslim, then there is no goodness in marrying her, and endeavour to take the children away from her by tricking her.
Allaah knows best.