It's intriguing that W gets all upset that her kids don't accept what she is doing. She is really in denial, thinking that her kids are 5 years old.
We are talking about big kids here, they get it all maybe much better then the adults involved. Your W needs help now, she will need to come to terms to understand that if she wants this new life then she will have consequences.
The kids don't need to accept her bad behavior much less have this idiot OM in their lives. Your W is replacing her family with some guy and wants everyone to bless her bad choices.
She has the right to say that she does not want to be married to you, but she has not right to impose the OM presence for the kids or you as a matter of fact.
There is something called "Respect" and she must learn the meaning of that word very fast or she will loose the little understanding she has from her kids.
Maybe there will be a way for you to tell her that it is not your responsibility to be bridge between her and the kids, it's not your job to make the kids understand that she is the mom and needs to be loved no matter what.
Why is this the WAS are so blind and so selfish to think that everyone needs to accept and understand everything they are doing even though they are the ones hurting their family.
Your W will have to understand that her choices and actions have serious consequences and she can't ignore the fact that her kids are hurting a lot right now, and will maybe hurt forever.
I know you can't blow on her, but she needs to get to terms of some responsibility of her actions. Maybe you too could go somewhere for dinner or so, and then you can express, very polite, no anger, all what she needs to understand about 4 teenagers and how it will impact their lives forever if there is not a clean path.
I hope she stops the crying guilt and start being a mother, but I also can see that she is total denial, she is just thinking about herself. You are the one with the hard load and yet in her eyes you are the one she can't come back to.
I will call on the vets to your sitch, it's time you get some better directions of what to do since it's involving a life of four innocent young people.
I can't say no more how I care for the little ones, I feel that even my R never gets better, I have an obligation to make minimal the suffering on my children.