Even if he was wrongfully accused, I highly doubt that because of his behavior.
1. He knows that he was convicted and "how that looks".. he also knows the laws about registering and keeping away from children. He's chosen to completely disregard all of that and disregard the path of being on the straight and narrow, in favor of getting with someone who has kids and has a lot of contact with kids. Wrongfully accused or not, at the very best it shows a serious lack of judgment to get with someone like your SIL when he has a legal history like he does. At the worst, it's him selecting a person who will give him close contact to victims he can groom to abuse.
2. He didn't tell her. When your DH called her, she wasn't aware.. the lack of responsiveness on her part is covering up the biggest red flag- she didn't even know. So it's not like he flat out told her, "I am a convicted sex offender, but I was wrongly accused and here's my story." He kept her in the dark and that is very suspicious, especially since they're to be getting married.
In my opinion you need to call the police and his parole officer and notify them of what's going on. Even if he's updated on his address, they may not be aware that he's around children. Pedophiles are court-ordered to stay a certain number of feet away from children and aren't allowed to live in the same house as a child.
I also agree with whoever said you should work on your relationship with the kids, just in case- you want to make sure they feel they have somewhere to turn if things go south quickly. It blows my mind that parents like your SIL exist- why would anyone put their innocent children in that kind of a situation??
Yes, yes, yes!
*warning SA mentioned, triggering*
My mother started going out with a convicted pedophile when I was 11. Everyone thought he was really charming, even my grandparents liked him. He was from another country and had obviously moved to escape being taken into custody again - I later found out he still had an outstanding warrant. OP, you mentioned your SIL is always around kids as well as having her own. These "people" groom their victims carefully to get access to as many potential victims as they can. In my case, I met the "man" (can't call them that really) at a school friend's birthday party. He was at the neighbor's, who was also throwing a birthday party. He hung around the girls' party all night. We had a chat. I must have mentioned that I kept snakes at home, because he turned up at my mom's house months later with a snake book in his hands. He'd asked my friend's parents for my address.
The rest of the story is rather painful. And my mother just did not see. You and I would probably call the police if a grown guy our daughter met turned up with any excuse, but he then pretended to be in love with my mom and she fell for it. I don't think I'll ever completely heal from what happened next. My mother not only ignored all the warning signals but told me I was lying when I informed her he raped me. He also took the opportunity to rape a class mate too. I wish I had someone like you around during those years, OP. Please be there for SIL's kids in anyway you can, but also - call the police, do whatever you can. Everyone at my school must have had their suspicions, but nobody ever helped me.
I'm surprised that so many poster say that these things are often made up and blown out of proportion, and that SIL's fiance might not be a pedophile. There is no reason to give them the benefit of the doubt. Behind them, they leave a path of destruction, hurt, and permanent damage.
I am so sorry this happened to you, and that nobody helped you. 
No child should have to experience sexual abuse and victimization, and nobody should have to live with the pain that results from being sexually abused.
ITA with your post, and especially that which I bolded.
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