What an absolute bellend.
I'm sorry that your sister, her DC and you are all going through this. She's very lucky to have you and your DH, supporting her through this.
Right now, her whole life has been turned upside down. She's had one vision of her life and that has now gone up in smoke. I don't blame her at all for being inconsolable and rage-filled. I would be, too, if I'd devoted decades of my life to raising children and keeping house and supporting my partner to build his wealth-only to be left for another woman.
Given how calculating he has been with his exit, ensuring that she benefits little (bar what he can't legally deny her) and he walks off with everything-I'm wondering if the abuse was just financial or if there wasn't some other coercive elements at play as well.
What she needs to do is what many other posters have said and it's going to be tough. She's going to need to find the strength for herself and her DC to get through it. I suspect having a plan and a focus may actually help her a lot.
First, she needs to book into see a GP and be honest about how she is feeling and coping. They may be able to give her a doctor's note that she can take to the job centre.
In the meantime, she needs to complete a claim for UC. I'm not sure what or how much she'll be entitled to-but she should be able to get something that'll go towards covering the basics.
Is the claim for Child Benefit in her name? If it is-and I'm not altogether sure about this as I've never claimed-but maybe there's a way to redirect the payment from his bank account to hers so that he doesn't receive it. It's at least worth looking into.
When she does go to the job centre, she needs to be honest. A lot of people have the view that the job centre staff are all nasty, judgemental people-but that is not the case. I've claimed JSA in the past-and found a lot of the advisers there very helpful and kind. They can, from what I remember, put people on courses for retraining if it will help get them back into work. This may be an avenue worth looking into as she needs a job and needs one quick.
Then, she just needs to take whatever job comes up. Anything that will help her to pay the bills. Put a claim in for CMS if her twat of an ex-dp refuses to pay his share.
As for the house, she needs to accept that it is going on sale and she needs to realise that, long-term, it is in her best interests to have it sold. It'll free her from as much contact with the bellend as possible and mean she won't have to face any legal fees if he does have to force a sale. In the end, it's inevitable and the more she kicks and screams and drags her feet-the harder it is going to be.
Hopefully, there's equity in the house and she can walk away with enough to start a new life.
She also needs to stop contacting the OW. It's not helping her. Her anger is understandable and I hope the OW realises what an utter dick she has saddled herself to-and feels a portion of what your DS feels. She deserves it, but right now, it's not helping her.
It could also be harming her. If her ex decided to try and push the "mentally unstable" line around to her kids and maybe even elsewhere to force contact or custody-he could end up with evidence of the instability sitting in his new partner's phone. She shouldn't just stop-she needs to.
One day- and it will seem like a far off day right now-she will look back on all this and realise what a dick he was and how much better off she is without him. It won't seem like it now-but when she gets back on her feet-and she will-and she has her independence and her kids and family around her-she may even realise that she doesn't want him back and that the OW is welcome to him (if he hasn't dumped on her by then).