My concern is with his statement.
To answer your specific question, however, if she files and is granted an annullment, the marriage is determined never to have been a sacramental marriage; and it applies to both parties as a sacramental marriage takes two people. In a marriage, one party cannot have a sacramental marriage and the other party not have one.
More specifically, you have raised a red flag. He appears to want to have nothing to do with the Cathoilic Church. You are talking about the possibility of marrying someone who wants nothing to do with the Church and at the same time talking about raising your children Catholic. To do so would be to raise children in a household where their father is at the minimum ambivalent about the Church, and possibly downright hostile. Either message that he sends to the children will be a direct contradiction to what you will be trying to teach them. It is difficult enough to raise children when both parents are on the “same page”. You are talking about a situation where you are not on the “same page”, and perhaps not even in the same book. Whether you like to hear it or not, you are inviting untold chaos and heartache.
If nothing else, you need to talk to some Catholics who are in similar situations; and I would include Catholics who are married to Christian non-Catholics of strong faith. They will tell you of the problems and heartaches they have encountered.
I would sugest some serious soul searching as to why you are dating someone who has said that they do not want to be married in the Catholic Church. You might want to talk with a marriage counselor, at least initially by yourself, to sort out what is really going on and driving you in this direction. You may also want to bring him into the counseling at some point, to find out what is driving him. I don’t know from your comments if he is Catholic or not; either way, you need to get to the bottom of several issues before you get married
If you don’t, you will get to the bottom of those issues after you are married, almost always with results not to you liking. You will either end up in a long and painful process of resolving the issues, and possibly giving up positions you now strongly hold to keep the marriage afloat, or not resolving the issues and living in the ambivalence of an unhappy marriage, or in divorce court. The issues are not going to go away. There is no magic, and ignoring them won’t solve them.
As an aside, unless you and he can find out and deal with why the first marriage didn’t work, you are very likely to have very similar problems. It is all too easy to blame the other spouse for the problems and the divorce, but it takes two to make the problems.