At mid-night of the second day* the parent will wake up and turn on the lights, but the baby will remain sleeping in bed. The parent will just stand there for about 3 minutes, then jump up and disappear off the screen. (The lights will turn off again, automatically.) When the baby wakes up in the morning, s/he will be alone.
* example: if your Tamagotchi character mated and had a baby at 10:30 AM Thursday, then the parent will leave the baby at midnight as Friday becomes Saturday.
Note: if you pause your Tamagotchi for long periods of time, it may take longer. Also, if you pause your Tamagotchi at night, or adjust the clock to skip past midnight, the parent won't leave when expected to.