This whole scenario is extremely unhealthy amd my advice would be to stay as far out of it as you can. The children are the only innocent party in this and they should not be being used in this way.
BIL is obviously a disgrace. To have done this to his children and his, and to put his brother in this position, and to seemingly still be going about being "offended" that you aren't all treating this like it's normal and OK just shows how self centered he is. Limiting your own children's contact with him seems a good idea. He is only out for his own interests.
The mother of these children is obviously using them as pawns. She wants to integrate herself into his extended family. Sending prezzies, demanding contact with your children, being the one to text SIL off his phone.... she isn't just some poor sap he's duped. He has obviously spun her a yarn, but she seems to know just what she's doing. She obviously won't rest until she's got BIL all to herself and is fully integrated into his family. She's using her children as pawns. No doubt when BIL and SIL's children are old enough to be on SM or something, if they make it that far without this coming out, she'll make it her business to tell them herself too. So all in all she doesn't seem like the sort of person you'd want your children around either. Her and BIL actually sound quite well matched TBH.
SIL was obviously the victim in this. Until she was told about the affair she was the innocent party, and what BIL has done to her is horrendous, it's good she's got so much support around her. However unfortunately it doesn't sound like she's really forgiven him (understandable, but important if she's going on with the relationship) or is handling this in a healthy way. That isn't a dig, it's understandable she wouldn't be reason personified in such a horrendous scenario, but it does seem like she hasn't really thought much beyond the present. When a spouse cheats you have two choices. Break up or try to forgive and move forwards. She has elected to stay in her marriage. But she's attached to that unrealistic conditions. No one would have blamed her for divorcing BIL, and what she's been put through is unforgivable. But, with there being children from both sides of this affair, it wasn't ever going to be as easy as brushing it all under the carpet. Her own children will eventually learn they have siblings. It's an open secret within the family already. Unfortunately SIL plan to keep it from them, and have another baby with BIL (which does seem like she was trying to make a point to the OW that he's still "hers" and she's got a younger baby with him that OW has) isn't sustainable. On top of that it seems the affair is still going on. The way all three parties are handling this (and again this isn't to blame SIL, this is on BIL, and sympathy is rightly for her) is a recipe for even further disaster.
It seems likely that OW will be pregnant again. BIL thinks he can have his cake and eat it. SIL is engaging in "pick me!" and trying to cement her broken marriage by clinging to BIL tighter and having another baby with him. The OW is on a mission to get BIL whatever the cost.
Keeping your children out of this ticking time bomb is all you can really do. DH can pick for himself whether he wants a relationship with his DN's, but the children should be kept well out of it. For a start it's totally wrong to make them complicit in this deception of their cousins, because when the cousins eventually learn they have siblings they are going to feel so betrayed by everyone (their father, their mother, the extended family....) and knowing even their (similarly aged?) cousins knew about their and hung out with their siblings will be too much.
Protect your own children.
Support SIL, and don't keep any more information from her.
Do not engage with the OW, because she sounds extremely manipulative amd us using her own children so won't think twice about using you.
Limit yours and your children's contact with BIL. DH can still see him but your children don't need embroiling in his unhealthy and lacking in guilt dynamics.