He deliberately set himself up as your victim and has made you the scapegoat for his wrongdoings. He did this as an advance alibi, for when he left. He'll (and his family will) eventually do the same to OW, but she's made her bed and that's not your problem.
You will need an excellent solicitor. Come to terms with the fact that child arrangements and finances will likely go though court. Prepare yourself for this, find acceptance. As the saying goes, when you're going through hell, keep going. You'll make it to the other side.
File for divorce on the basis of unreasonable behaviour (including the slander to family and friends and cheating and leaving for/living with other woman. Filing for adultery requires him to sign a confession statement, this one won't so don't waste your time.
As part of filing, file for child arrangements and financial matters to be determined. You can still negotiate and reach a settlement before court, this just ensures you dont needlessly wait and waste time if he has no intention to negotiate in good faith.
Finally, dont allow yourself to be railroaded into accepting 50/50 childcare if that was not the arrangement during the marriage. Changing this now won't be in the best interests of the children at such a traumatic time of family upheaval - but would serve to advantage him in financial proceedings (and would mean he pays zero child maintenance).
Ensure you are claiming child benefit for both children in your own name.
Be clever and trust nothing he says. Only trust the final court approved consent order.