You've received a lot of very good and mature advice on how to deal with this very potentially sticky situation, and clearly you've given this a great amount of thought...much more so than most people nowadays in their dealings with others. You are to be commended for your thoughtfulness.
It sounds like your biggest concern, other than being stuck between them venting back and forth about the other one, is what would occur in your relationship with her ex/your boyfriend if he were to realize that you knew all along that she had saddled up another horse and was riding him about while her then-husband was still tethered up in the barn, so to speak. Is that a fair assessment?
I mean, you didn't seem concerned how the kids would take the news that you were all moving in together, but then again the kids have probably been friends for years and they were probably all for it. You were a little concerned how YOUR ex would take it, but it sounds like he took it like a champ, kudos to him. Having been the child of multiple divorces, I can tell you from personal experience that when the grown-ups act maturely and don't degenerate the situation into backbiting and finger-pointing, things work out for the best for all parties involved. And given that HIS ex and YOUR ex are still deeply involved in the picture and everyone has children together, well, clearly it's in everyone's best interests to handle this with the utmost maturity, even if you didn't want to keep the best friendship with her or a civil relationship with your ex.
Your best friend is simultaneously a little sad to see you so happy with HER ex, wishing a little bit that she hadn't bailed on the marriage (yeah, she's probably more happy FOR you being happy with him, but a little sad nonetheless, that's just human nature), as well as apprehensive that at some point you're gonna spill the beans on the true nature of the demise of her marriage. And there's probably a part of her that wonders how long you've had designs on her ex-husband, whether or not those predated the end of her marriage, etc. There's a lot of conflicting emotions going through her right now, and as I type this I can't recollect if you've mentioned whether or not she's still married, or if that current or former marriage is/was happy, what-have-you. Even if she's blissfully happy, she might still be a tad ambivalent that you're happily dating her ex-husband, a guy you've known for years. Again, that's just human nature.
You should tell her that you want her to be happy, but you also want YOU to be happy, and you're very happy with her ex and you'd still like to be able to be best friends with her. That being said, you'd like her to come clean with her ex for the following reasons...(1) if he ever finds about the circumstances under which she ended the marriage, the best friend will IMMEDIATELY assume that you told him all of the juicy details and no matter how much you protest your innocence you'll never convince her of it, (2) he'll be so pissed that you kept this from him that he might bail on your relationship and upend your life, (3) the marriage had been over for some time, both in legality as well as emotionally and everyone's a grown-up and the truth should at last be told, and (4) as other posters have mentioned, the poor guy might be driving himself a little crazy NOT trying to do some of the same things that he erroneously believed doomed his first marriage, and you'll go a little crazy yourself knowing that he's doing things and acting in a manner to prevent the same thing from happening again when you have the means to prevent that, and the blame for that can be laid at the feet of your best friend. Additionally, you should probably be present when this conversation between exes takes place, as she might downplay her role in the demise of their marriage, as well as eliminate the necessity of him asking you what elements of her story coincided with what she told you. Sometimes "truth" is subjective, and while you would want her to be completely forthcoming in that revelation, she might choose to keep certain truths to herself that he would need to know...your presence for that conversation would be more persuasive in turning subjective truth into factual truth.
As to his reaction...most guys might have pretty strong emotions about it at first (he'll be pissed), but after he thinks about it for awhile he'll probably mellow out. He's going to initially wonder what you knew and when you knew it, he'll be irked that it would appear you let him flounder about in a bad marriage and a messy divorce (later on, he'll realize that you were stuck between a rock and a hard place in terms of your best friendship with her vis-a-vis your friendship with him)...but if he can be made to see that you didn't believe it was your place to tell him then (it was his ex-wife's responsibility) but now that you two are building a future together he should know now so he can move forward and not beat himself up about the previous marriage or spend too much time on autocorrecting what wasn't really broken, then he'll see you as someone he can rely upon to trust and keep secrets. Additionally, once this particular cat is out of the bag, after some introspection he'll probably feel a little better knowing it was HER and not HIM who had the marital issues (again, probably an oversimplification on my part, not knowing what the particular issues were...maybe it WAS him, how he acted or did not act that drove her to another guy, I don't know), but lI believe one of the above posters said that he did feel better afterward once everything was out in the open that it wasn't him with the problems, it was her.
At the end of the day, it's all upon your best friend. It was her responsibility to tell him the truth then, it's her responsibility to tell him now (OR allow YOU to tell him the truth now, so it's out of the bag and EVERYONE can move forward, and if she truly wants to be your friend then she needs to rid herself of any lingering resentment on her part and let you be happy with him. All you're doing is holding up a mirror...if she doesn't like what's reflected, then it's up to her to change it.
This is not to say that all four of the grown-ups are going to live happily ever after...but clearing this air with them makes it a damn sight more likely that friendships will be salvaged.
You've gotten a lot of good advice and food for thought. I think I can speak for everyone here when I say that we wish you the best with this situation, and that your heart is in the right place. We can only wish that all people were so considerate.
Keep us posted.