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 mother of these two children embraces Islam, then they follow the religion of their mother because the children follow the parent who has the best religion and they are ruled to be Muslims; for more benefit in this regard, please refer to Fatwa 81957.
As regards their fostering, it is the right of their mother, and she should be keen on nurturing them according to the creed and morals of Islam. Nonetheless, their father has the right to see them and visit them, however, she is not allowed to allow him to enter upon her or be in seclusion with her, and she should not allow him to influence them.
On the other hand, you did not mention whether or not that man divorced her; if she is still married to him, then if she embraces Islam, their marriage becomes void, but if he embraces Islam while she is still in her waiting period, then they are still considered as a husband and wife with the first marriage. However, if he does not embrace Islam until her waiting period finishes, then she is permitted to marry another man after this; for more benefit, please refer to Fatwa 84226.
Even if this woman embraces Islam, we do not advise you to marry her until it becomes clear to you that she performs the rites and obligations of Islam, and until you perform the Istikhaarah prayer about marrying her.
For more benefit on the Istikhaarah prayer, please refer to Fatwa 81434.
Allaah Knows best.