I absolutely agree that this boyfriend (definitely not partner - 10 months in, don't live together, already starting to display dominant/controlling/abusive tendencies to young children and their mother...? Not partner material in the slightest!) needs to be thrown to the curb, @candragonsbepurple, and with immediate effect.
However, it's not that grown men ought not to turn to their mothers for advice, or even just a bewildered "where did I go/what did I do wrong...?!" conversation. That isn't necessarily the issue. The issue is that the boyfriend's mother (not the MIL - partly because, hopefully, there's nothing in law connecting the two of you!) got involved. Took it upon herself to berate you, and then essentially belittle your parenting choices - especially when by doing so, she was displaying how poor her own are!
I mean, my DS (19) knows that he can turn to me and talk to/with me about absolutely anything. In the last few days, I have counselled him about a relationship issue, peer pressure problems, and heard far more than I want to about how his driving lessons are going. Last week, it was another relationship issue that looked like turning into a crisis if he didn't put a boundary in place. At no time during any of these conversations - and I'll admit, my maternal/protective streak is pretty strong, and I am still seething quietly about last week's - did I think to myself: "huh. I'm going to fire off an 'e'mail essentially telling DS' girlfriend what I think about her latest issue with him...". Partly because they are both adults, they are both at fault, I am merely a sounding board that has the capacity to tell my son when his head's up his bloody arse!
Again, there are faults on both sides here - but there should only ever be two people in a relationship. In your situation, OP, there's you, the (hopefully very discarded) boyfriend and his mother. Ten months in, and you can already tell how much worse it'd be two years in, five years in, ten years in...
And not only for you - but for your kids, too.
I think your ex-husband is commendable for having found a way of communicating with your children where he probably knows more about their lives than he would if they weren't distracted by the online games. At the very least, having put their educations and burgeoning friendship groups before his own needs as a parent, he is still present in their lives. He probably could tell you other instances where your boyfriend has undermined your authority as the actual parent and perhaps made your son, at the very least, feel like shit. Because this, I'd stake my life, won't be the first time. Low level abuse ramps up into shit like this "my roof" and "your kids are rude brats: do better" bullshit, And then it ramps up again. Also, at the end of the day, your ex is losing his father, but your kids are losing a grandparent - and possibly the first one that they know (as in the first death). When you're the ages your children are, and your Dad moves miles away, and your grandparent is dying - and then suddenly you've got some cocklodger who's been on the scene for less than a year throwing his weight around in your family's home... Yeah, that's a scary emotional ride for them.
Fair enough, you never moved him in with your kids - but you were contemplating it if he's already been introduced to them (and far too early, unless there's a drip feed where they've known him their whole lives...? In which case - doesn't matter; still far too soon). I really hope you don't continue any sort of a relationship with you, because he (and his mother) really have told/shown you who they are with this escapade.
Listen to/believe them the first time.
(Edited to add that I would tell the mother precisely what she could do with her parenting tips and how she's failed in raising a man, rather than an entitled and arrogant cocklodger... I wouldn't be tempted to do so. I would. But I detest women like this who give the ones who listen to and advise our sons, smile nicely at their awful girlfriends/wives... and keep our mouths shut and our fingers off the keyboards, a bad reputation.)