Please ignore all suggestions to play games/withhold contact from his children/tell the woman how you were sleeping together all along.
Firstly, it’s not up to you to decide who he introduces his children to, because they are as much his children as they are yours, so while you may not like it You’re not together any more, and he has the right to parent as he sees fit, just as much as you do. And the courts won’t look on you favourably, and no, six month court delays is not an excuse to use the children as pawns.
Secondly, while there is every chance he was with this woman before you split, there’s also a chance that he wasn’t. I know at least two couples who got together fairly soon after a split, and they were accused of having an affair, and I’ve read posters say on here that “If someone appears on the scene within a year they were obviously the OW,” wen this so very clearly is not always the case.
Yes, they may have been sleeping together, but equally he may have developed feelings for her and left anyway and they got together afterwards. Either way, if you start talking about how you slept together until the end etc you will look a bit desperate and pathetic. Don’t lower yourself to that. You’re not together any more, move forward with your own life, and be glad you’re no longer a part of his.