Know your limits. If your wife no longer wants to be wife to you, you can't force it. She left a Godly family for her reasons and perhaps she had this bf all along. Adultery is grounds for divorce in scripture. You are entitled to be happy, as is she.
You loved one another once and still do since you're parents. Consider that for the children, part amicably and stay friends. And of course pray.
Sometimes God lets our mate leave because they're no longer made for us and both people in that relationship deserve happiness. It does a disservice to all the years of love between you to part as enemies. And it sends a terrible deeper heartbreaking message to your children.
As for the other man. If it were me, first I'd have to surprise my husband by turning into a man, ha! OK, trying to make you grin just a bit there. Did it work?
Seriously, if it were me, I'd do a background check on the guy. You have a right to know who is going to be with your wife and children.
Very likely, she didn't do this.
There are things you're going to have to accept here. First, you had nothing to do with her decision to leave. It isn't, wasn't, your fault! It was her choice.
Secondly, you don't know if that other guy even knew she was married when they first met. Don't blame him. Your wife is responsible for all of this in the leaving part.
It takes two people to make a marriage crumble. And if it be Gods will, two people to heal it too. Don't push. She made her choice, let her see how it is to live with it. But be there for her if it all goes south. And as I mentioned, be amicable. Don't hate. Hate destroys the one who feels the hatred, it does nothing to the target of the hate. I learned that from my sister and her nightmare of a marriage to the wrong man.
God be with you all.