I am sorry for your husband's loss. So in this situation it does seem like a lot of the damage is already done. No one can rewind time and have the situation go down differently. However, if he feels up for it, he may want to address things with his sister. He ultimately can't tell her who she can and can't be friends with, and as inappropriate as I think it is, sometimes these relationships continue between exes and family, But he can ask her to have a boundary about family information and her friendship with his ex.

Death brings out the worst in people. It was out of line on his sister's part to inform the ex before her brother. She is effectively excluding her own brother from his family and disrupting his mourning process by involving the ex. She was so inconsiderate. 

Just be a shoulder and an ear for your husband right now, it's all you can do.

Thats horrible - SIL sounds very toxic.

Condolences to your family. I guess you can expect BM to be there at life celebration too...

To remain besties with a high conflict gubm after a divorce because for whatever stupid reason, they want to remain friends. Your husband/partner has addressed with his sister before but he cannot control her relationship with the ex.

my own husband's sil is still besties with his hcgubm exwife, they are the same kind, would rather break up relationships and steal men, gold diggers etc. On our wedding day in fact, hubbys sil called the exwife to tell them she went to our wedding and i wonder how long this marriage will last. Normal people don't do this but they are toxic people and toxic people surround themselves with toxic people. They love the drama and to hit back

your husband has voiced multiple times how much he doesn't like this relationship his sister has with his ex, especially if that relationship was high conflict and abusive, its hard for any sane individual to want to maintain a relationship but sil chose to. 
 

all your partner can do is disengage from her and keep contact to what is necessary. If she actually cared, she'd know her place and what is appropriate or not.

Does BM have primary custody?  If so it makes sense that the kids would find out on her end before DH could tell them. I doubt DH will be able to stop the flow of information between those two. Although it may be annoying to you and DH there's nothing either of you can do about his sister's close friendship with his ex-wife. Marriage creates bonds and brings a couple and families together, divorce dissolves a marriage but SIL is still an aunt to the ex's kids and obviously her friend.  My DH is still friends with most of BM's family. I used to work with one of BM's brothers and he and I became friends. He would tell people that we were family "in a roundabout way."  Serial monogamy complicates relationships!

Time for a scorched earth policy with the X and the sister.  No more tolerance, nothing but "No" and offensive and insulting references toward the X with the sister.  
 

The kids have to see and recognize BM's crap so it is time for them to see their father be a confident assertive zero tolerance man with BM and the aunt.  Of course the insulting references regarding the X do not occur around the kids.  He should refer to her as "your mother" or by her first name when speaking g of her to the kids.

He has to learn to not be hurt by their crap and return nothing but insult and public humiliation.