I think you need to be very clear about how far you actually want to apologise, how far you want the children to be in contact, and how much you want this to be a 'lifeline' to your rattish exP.
HE was the one with responsibility to his wife and family, and as is often the case, a man who treats his wife with no respect also treats other women he involves with no respect either.
I think you should take this in very separate stages, and see each stage as indepnedent of the others - i.e an apology to the wid=fe is for and of itself, not a good stepping stone to contact between the children.
And be careful - this 'children asking for each iother' sounds a bit odd to me - is he trying to mastermind a little brood of his various children, are you all using it as a way to keep him involved? How do these children know so much of each other? Not saying it's a bad thing, but ...
Stage one - tell the ex-wife that you do feel sorry and guilty for the affair, that you knew he was married and had children, and you regret that you went into it. Do not offer the (very sad) circumstances you were living under as a qualification or excuse, if you allude to it at all, admit that despite knowing all that, you succumbed to an affair, and even though you were lonely etc, you wish you had not done it.
Don't mention the children getting together in your apology message. Leave it and see if she replies, and what she says.
Your lovely child's father really is a rat. He doubtless preyed on your vulnerability - but then completely failed to look after you. You sound lovely - and I hope all this works out well and happily in the long run.