I can only imagine the pain and frustration you are experiencing as a result of your W's actions. I will tell you from the viewpoint of the wayward W that she feels she doesn't have to give you an account for her how she lives her life......which includes parenting. She feels every time you confront her with something concerning her and the children, it's none of your business (to be frank). I empathize with you. I was a very close observer when one of my adult children experienced the same situation where the OM was introduced three days after the H was kicked out. I saw the raw pain it caused. I saw my grandchildren in a state of hurt, fear, and confusion b/c they were given no time to even realize their parents had split, until the OM was moving into their home. Here's the thing, unless the law is broken or child services has a legit reason to step in, there's really very little you can do directly to prevent your XW from doing anything she wants to do.
Will she be selfish? Of course! She's a WW and nothing has changed. Will she be unfair to the children? Of course! She's a WW and her mindset has not changed. Will it do any good to point out the mistakes she makes with the children, or how they are being affecting? Absolutely not! Look, she doesn't care how you (or the kids) really feel about any of this. Oh, she may say something to "sound" as if it's for the best, but bottom line.........it's still all about her.
I think this will be harder for you than anything you've had previously. No matter how much you protest, she is still wayward and won't win the best mother of the year award. She won't even take simple, rational steps that requires her to be honest with her children, or to put their mental health above her own desires. And as far as you are feeling she should have discussed things with you so you'd be aware and know how to answer any questions that may arise.........let it go. I'm not saying you are being totally unreasonable, but she doesn't feel she has to comply.
It's very, very tough. You can't make her be the mother you want for your children. She'll do her thing, and you'll have to do yours. It's not fair to the kids, but there's nothing you can do but focus on your own parenting. I really understand how it feels when the kids ask questions about the other parent and the affair person. My grandchildren would ask me, and I was determined I would not talk bad about their other parent, but neither was I going to lie. It's hard to answer their questions, especially with very young children. Don't make excuses for their mother's actions, and be honest with your children. Try your best to keep it age appropriate. One more thing......it's okay to answer with "I don't know".
It is not about what you feel should work in your M. It is about doing the work that gets the right results. Do what works!