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 current situation of that woman is that she does not believe in any religion, then she has indeed apostatized from Islam and her marriage is nullified. No consideration is given to what you mentioned of her having a lot of knowledge about Islam, or that she agrees to give your children an Islamic education and so forth. Hence, you must separate from her and it is not permissible for you to have sexual intercourse with her; if you do, you would be considered committing adultery. For more benefit, please refer to Fataawa 5985 and 134052.
On the other hand, it is obligatory to call her to repent and return to Islam, and to explain to her the danger of apostasy from Islam. You could perhaps seek assistance in this regard from some people of knowledge. If she does return to Islam while she is still in her ‘Iddah (waiting period), she would return to you with no need for a new marriage contract. But if her ‘Iddah has finished, she cannot return to you except with a new contract.
Anyway, if she insists on remaining an apostate, you must separate from her and there is no need for divorce, since the marriage was nullified by her apostasy. The fact that the marriage was initiated in a legitimate way does not mean it remains such after apostasy.
Kindly see Fataawa 55132 and 87672 about how to call an apostate back to the true religion.
Finally, the disbelieving woman has no right to custody of the children, so try your best to attain custody. If you are able to take them, even without her knowledge, and travel to reside with them in a Muslim country, then do so. This will help you in raising them upon the creed and morals of Islam.
Allaah Knows best.