I have two kids, and work 5-6 nights a week. My current schedule gives me a lot of time with my family, but it has me feeling like a zombie a lot, I just deal with it.
The first few weeks after birth are the most difficult, especially with your first baby, so I would suggest making that the priority until your wife and you are adjusted.
Currently I train 3-4 days a week. M/W/F with the occasional Sat. My wife understands that I need BJJ to stay sane, and she is very supportive. Tues/Thurs I am with my family all day, and the weekend is all for my family as well. I pretty much will do whatever my wife wants on my days off from BJJ. Any chores etc, and if there aren't any, its a few hours at the park playing.
When I go to a Saturday class, I take my son with me, and then we go to the park for the rest of the day. I bring some toys with me, and he plays with them there. He isn't that interested in BJJ yet, but hopefully soon he will be.
Right now he loves the soccer ball more than anything.
Don't get me wrong I wish I could train 5-6 days a week, but unfortunately this is my life and I accept it for what it is. My father wasn't around too much when I was a kid, he worked extremely long hours, so for me it is very important to spend a lot of time with my son and to teach him what I can. My daughter is still in larvae stage, but once she can walk she will be right there with me and my son.
You only get one chance to raise them right. You have the rest of your life to train BJJ.