Quick update - I've tried a couple of scenarios, just wanted to add the info that none of them worked 🙂 I am playing a legacy challenge with base game and freebies only. The first time this happened, the mother was in the tech guru career and the father was unemployed, but got a job during her third trimester. He had not started work before the birth. Pregnancies 2 through 4, both had jobs; he was a Writer in the Author branch and she was a Tech Guru, both had vacation days available during the pregnancies and births.
Next generation, in the same save game, mother was an Entertainer in the Musician career and father is unemployed / stay-at-home dad. She had twins the first time and a son the second time. Vacation days available for her on both pregnancies.
These are the choices I've tried:
Do Nothing - the working sims will be sent to work every day. This is the "option" I used most often. The baby was never born at work, but I had one that was apparently in labor at work and "gave birth" on the lawn right after work. The others usually came a few hours before or after work, which probably coincides with whenever they 'woo-hooed' to get pregnant.
Take Vacation Days - there appeared to be no penalty to progress if you take 2 vacation days in a row after the birth.
Cancel "Go to Work" - this last pregnancy, I tried cancelling the go to work task every day after she gave birth. The job button was flashing red during the entire work time but I just ignored it. I had hoped she was already marked as on family leave, with the bug being in the autonomous part. She didn't get paid, and on the 2nd day she took a hit on her job progress as well. The bar was on Good, almost to Excellent. No hit the first day, but on the second the bar dropped one tick to just above good. I decided to send her to work the third day, so as not to lose any more progress. After she worked on the third day, the message that "no more family leave is available" came up. It took a couple more days of doing the daily task to get her progress back up.
So, for me, the best option has been to either take vacation days or go to work normally. I hate losing the vacation days when I know I should be getting family leave, so most of the time I just let them go to work as usual. This is really only possible because I have a full household, so unemployed dad and various aunts and uncles take care of the baby. Mom's needs decay so much faster at work during the time immediately after birth, it's difficult to keep up with her schedule (just like in real life).