You are hurt and angry and that is understandable.
Ask yourself this though. If you introduced your children to a new friend (male or female) would you expect your ex to be informed.
If he introduced your dc to a work colleague, or other friend they had not met before, would he need to inform you each and every time? No because that would be plain silly.
So the issue really is that he introduced the 'OW'.
If, as he claims, it is not serious, then it is just like introducing another friend is it not? And therefore nothing to get concerned about.
But as you rightly point out he is probably lying to you and to himself about it all. You just have to accept that he will (as will a lot of men), lie about things to keep the peace and to make things 'easier' for themselves everyone. He had an affair and lied to you about your relationship, and you think you can trust him to do the right thing now? Don't listen to what he is saying, look at what he is actually doing.
He has moved out after an affair, he is introducing the children to the OW and he is lying to you about various things and has no empathy or consideration of your feelings or state of mind. Of course it is serious with her!
As far as the OW goes don't judge her without knowing. I was the OW once. The man involved wasn't married but had a child with his partner. He told me they had been separated for over a year and that he saw his dd regularly. He introduced me to his family (parents and brother) the same weekend I met his dd at his house, which was quite soon after we met but we had been talking to each other for weeks before we met (we met on an online dating site). I was happy for him to meet my ds, I didn't see why there was a reason I shouldn't meet his dd in the circumstances he had given me and that we were quite seriously in a relationship very quickly after meeting having had several weeks of online and phone calls before that. Plus he didn't give any reason there might be an issue.
His ex went crazy when she found out, stopped all contact with dd and embarked on a hate campaign and length court process. It transpired later that the 'over a year' separation was 'in his mind, I always intended to leave when dd was a bit older' (she was 18 months at the time) and he had in fact been living with her up to and including the weekend I finally agreed to meet him in person. He also admitted he didn't tell me any of that to start with because he knew I wouldn't of agreed to meet him. How right he was! But by then I was suckered in, was supporting him through the court process and my ds was very attached to him.
In hind sight his ex's behaviour was completely understandable, but not at the time when I believed what he had told me and had no reason to doubt him. I still can't believe his parents didn't say something at the time!
In your case I would imagine the OW thinks things are very serious between them. He has afterall left his wife and dc for her. He is rushing through the divorce for her. He is protecting her by keeping her name out of things. Why would she have any concerns about meeting his children when their relationship must appear to her to be very serious when he has done all that for her.
At the end of the day you can not force your ex to do what you want him to do. Just becaue you want him to wait to introduce the children to OW doesn't mean he will, no matter what your reasons for it he sees things completely differently. In the same way you can not make him be honest with you, or to consider your feelings. The best you can do is deal with the fallout from it and try your best not to influence your dc in their opionion of her (hard I know).
Things won't start to be any better till you stop trying to get him to do the right thing and simply accept he is going to do exactly as he wants pretty much regardless of what you say or what is best for other people as you see it. He sees things differently and is probably only agreeing with you for a bit of peace. Afterall, if he had turned around and said 'no i'm not waiting to introduce them to her' when you discussed it together he could pretty much expect you to fly of the handle and name her in the divorce etc. But by saying, of course I will wait, he has bought himself some time and less arguments.
It's a terrible situation and I do feel for you, really I do. But stop believing what he says, stop worrying about him, and simply do what is right for you and the children at this time. I would say rushing through the divorce when you're clearly not able to make any serious long term decisions is not a good thing. Some time to let things settle and to make sure you get the best deal for yourself and dc may be wiser. Although I completely understand why you want to be legally separated from this man given what he is putting you though.
Be kind to yourself, don't run before you can walk, and stop believing him to be a decent and honourable man! He's not.