I went through all this, before, during and after the relationship. I stayed for so many years trapped because of it.
When we ended he made so many threats and did make some attempts all to try to get back. This might sound awful but as people told me to try to get me to detach..it's not that hard to kill yourself, no one has that many failed attempts.
There was even a day he told me he had written a letter to the kids to say he killed himself because of me, then went AWOL, I was fully convinced that day that I would have to tell the kids he was dead.
It all kept me completely trapped, trying to look after him, but I had finally found the strength to say it was over and stick to it so I knew that I couldn't ever go back then or it would be another decade or more in a relationship I didn't want to be in.
One day after around 6 months I finally detached completely in my head and said enough was enough, funnily enough I found out then the day before he had been sent to a psychiatric hospital after another attempt.
I'm not saying this to scare you. I'm saying it so that you are a person with your own life to live, you can't get back with someone you don't want to be with no matter what they are threatening.
As soon as my ex got out of hospital and realised how detached I was from him he stopped all of that!! I was enabling it but letting him make the threats to me.
My best advice is to say to the mother that you've sought professional advice about this (she doesn't need to know that you haven't), tell her it is not in his best interests for you to take him back or for you to console or help him through these feelings, it's up to her as a mother and any professionals she thinks he may need, so you understand that she is concerned and probably contacting you out of desperation, but that is unfair, harmful to both of you and just passing responsibility onto you. Tell her your focus right now is on the children and that's where you need to divert all of your energy, instead of looking after your ex.