I'm so sorry to read this - I had similar happen to me, years ago, but at least there were no children involved in either situation.
What I learnt from it is that they start off feeling guilty but they can't cope with that for long, so very quickly the story will change and it will become your fault - this is due to them (both) rewriting history to make themselves feel better about their cheating.
You must not listen to anything either of them say that blames you, or her H, in any way - because it's only to make them feel better, which makes you feel worse. The only thing you need to learn is that he simply Did Not Love You Enough. That's on him, not you.
Keep contact to a minimum, and keep it practical. DON'T ask him what went wrong, what she has that you don't, or anything else - because whatever he tells you will be lies - guilt-reducing lies.
Get finances sorted ASAP while the guilt is still there - he may promise all sorts initially, but then will retract it over the next few weeks, as the guilt wears off and he moves on. Get onto a solicitor immediately and start the process.
If you can manage to keep the house, so much the better - but if you can't manage the mortgage by yourself then I guess it will have to be sold. Check how much child maintenance you should be paid, don't let him offer less than that, and don't believe any sob stories about him not being able to afford it. Too fucking bad, mate!
Her - well, I totally understand your feelings about the kids not seeing her, but sadly you can't control that. Initially you might be able to, in terms of only letting him see the kids at your house - but then YOU'd have to see him as well, and that's not in your best interests. So much though I hate to tell you this, you're going to have to come to terms with this - he is their father and he can take them to see whom he likes, so long as they're no danger to the kids.
Her DH must be in a state of shock too - but I'd avoid having much to do with him, even if he wants to - it's not healthy for either of you.
Tell friends, tell family - get all the support you need, let it all out - there is no shame attached to you, this is all on HIM (and your "friend"). And of course you'll have to cancel all the wedding plans - and trust me, you will have to do it, because (from experience) if you try to get him to, you'll be left in the lurch. I well remember getting a phonecall from one of my ex's friends, very hurt because he hadn't received his wedding invitation - and having to tell him that the reason why was because his friend [my ex] had fucked off with a secretary from work and there was no wedding. The friend was mortified - but of course the feckless ex hadn't bothered to tell him, dick that he was.
Focus on your children, keeping them safe and in a routine - it will help - but make sure you have people who you can call and let it all out to, because that will help too.
((((hugs)))) - it fucking sucks, it really does. 