I haven't voted, OP, because in one way you are totally justified in your anger and pain, and in another he has to be allowed to grieve his way and get on with his life because the kids rely on him totally now.
I think the raw pain of your sister's loss is now having salt rubbed in the wound with this woman.
But you don't know how established she will be, or how long she will last. It's very rebound for your brother in law. I'd be super fucked off with him too, though.
I think it's because he's brought her so quickly into the home.
Oh, OP, I'm so sorry - just putting myself for a moment in your position makes me want to go and get the kids and bring them to you and let him get on with whatever he's doing with this such different person.
Maybe you should actually have the kids with you as much as you can. Forget about him and whatever he's doing to get through. Just make your focus having some lovely times with them. Hug them. See her in them. She's alive in them. Without being in any way emotionally heavy-handed with them (you have to be very careful about this, because you're full of emotion and love), just have the best time with them.
And then he'll have time to do what he wants/needs to do.
Be forgiving, too. He needs to get through this. He needs to work this out for himself. And be the best parent he can be.