Morning OP, hope you've had a good night's sleep and a chance to reflect.
Pippop's is a good suggestion.
I'd be very wary of contacting her ex behind her back TBH, this doesn't seem particularly respectful of her boundaries. Obviously if you thought the children were being seriously mistreated it would be different.
If I have this right, what we know about the children is that (a) they are allowed to play computer games you don't approve of (b) she is a bit impatient with them these days and (c) the new man is being a twat and slagging off their dad in front of them. Oh, plus (d) their mum went out dating when she broke up with her partner rather than "concentrating on the children" in the way that you clearly think would have been preferable.
As far as I can see (c) is the only unequivocally troubling thing here, and sadly we can't do anything about this new guy being a twat (yet!). When to start seeing people again after a breakup, and where to draw boundaries for children, these are matters of individual autonomy, and you are entitled to give your view but ultimately they are her decision. The impatience thing may or may not be an indicator of an underlying problem with the new man. It could also reflect the fact that she is adjusting to parenting on her own, or a tricky stage in their emotional development. The trouble is we don't know the counterfactual - what would she have been like with them at this stage if it hadn't been for the new man. It could be that the man is a symptom of her feeling down and overwhelmed, rather than the cause.
I completely accept your feeling that she has changed, and is going through a rough patch of some sort, and I agree this man doesn't sound like great news. Nobody is better placed to know these things than you. I believe you. I would just be very wary of dressing that feeling up in any particular clothes, if you see what I mean, particular if it might in some lights make you look a bit judgey. The computer game thing is a case in point. If I had kids and my mother tried to say that to me I'd probably argue the case for the hell of it, even if I slightly agreed with her, because I'd see it as an aspersion on my parenting and feel defensive and embarrassed. If I was depressed and not thinking very clearly, it might also make me see her as "the enemy" and I'd become even more wary. I wouldn't have the spare energy to calculate that she was probably doing it as part of a wider pattern of worrying and caring about me.
Rather than picking on individual things that you think are wrong, I would try to stick to that wider pattern, and if possible talk to her about the general sense you get that she is unhappy. Nobody can argue with you about what YOU feel and your fears for someone else's wellbeing, whereas they can easily argue with you over a computer game, or the timing of a few dates.
If she just isn't responsive at all, then all you can do is keep in as much contact as she allows, and wait for her to come to you.