I am a late to your thread OP, but I am so sorry you have been treated this way.
He is a piece of work isn't he. It didn't work out with his side piece (no surprise there) and so now I will bet my bottom dollar he would very much like things to go back to normal.
You sound very wise, and you must remember that now the ball is is your court. Yes, it is very fucking unfair that it is this way, and that you have to now make the decision of how to move forward. But, saying that, you can do this however you please. And what suits you best.
A lot of posters will tell you to get him to move out right now, but if this doesn't suit you right now, then let him stay. And do his share of the parenting. On your terms though!
Perhaps something like he sleeps in the room with the baby and does the night wake ups. And all of the harder jobs that you are not able to do alone right now. Use him. As much as he has been using you. And then during this time, you decide for yourself how to want to go forward. You do not need to make a decision right now. But you have control of the situation in your home.
You can see how it goes when you start back at work, and once you have settled back in, you think about a new routine and how you are able to manage it. Call on family, let them know. You don't have to keep his dirty little secret. I know when my ex had affairs, I felt so embarrassed to tell anyone, as somehow I felt a a failure. In hindsight now, I see that is such a skewed view as I, like you, had absolutely no responsibility for his actions. It was a decision he alone made.
Whatever any one else advises, you must do what is best for you right now in this moment.