First of all I want to say that this may not work for all pilots/families/aviation careers. My husband was a fairly senior First Officer at a regional airline when my daughter was born so he had more of an opportunity to change his schedule around than I know a lot of pilots do.
When we went to our first doctor’s appointment our doctor gave us the “due date” of our child and we started with that date in mind. In my husband’s contract he is given 2 weeks off each year, my husband saved both of his vacation weeks for around that date. He bid for the 2 weeks surrounding the due date at first, just in case she came a little early or late. When the date got closer we found out our daughter was breach and with our doctor’s help we made the decision to have a scheduled c-section. When my husband heard that we would know the exact date of our daughter’s birth he was able to trade his vacation into the week of and the week after so that he would be able to spend the most time with us. We were lucky that she did not come 2 weeks before (not only was he working that whole time but his parents and my parents were all out of the country). I am so thankful that everything went according to plan, we woke up the day of her scheduled c-section and I got to have my husband by my side while our baby girl came into this world! I pray that the same can be said for our second.