How do I deliver a baby?

Solution 1:

Mating: Bring a male and a female Dweller together into the living quarters. In larger living quarters you can bring in more Dwellers and let them work it out. The higher the Charisma the faster they will mate.

Direct offsprings will not mate, the game prevents incestuous behavior. If you do not keep log of your bloodlines (which you most probably don't) there are two ways to detect blood relationship:

1. By default new borns will get the last name of the father. So if two dwellers have the same last name, they might be related.
2. In case related dwellers are in the living quarters, they will bring up the topic that it is nice to meet with the family.

Pregnancy: While women are pregnant they can work. But they will not take care of any incidents but run into the living quarters instead. Make sure to assign at least one male/non-pregnant Dweller to all stations and don't equip pregnant women with weapons since they don't use them.

Deliver Baby: After mating it takes 3 hours for a baby to be born. You do not have to be in the game for this whole time. Just come back after 3 hours and you'll have a baby.

Make sure to have enough room in your living quarters. If your vault is full, women will simply stay pregnant until there is free room. There won't be an indicator and neither the pregnant dwellers will complain in any way. (Also see maximum population below)

Growing up: It takes another 3 hours until children are fully grown up. In their stats they have a grow-up-bar. During this time they can not be assigned to jobs. Again you do not have to be in the game during this time.

Overpopulation: Take care to not grow too fast. Once a baby is born you will need to feed it. Since children won't work you quickly run out of resources and have to juggle your workforce between your water treatment plants and diners if you get too many babies at once.

Maximum Population: The maximum number of dwellers in the vault is 200 - if you reach 200 dwellers, you have to be very careful placing men and women together in living quarters, or you'll have a whole lot of perpetually-pregnant females.

Solution 2:

You need to wait three hours of game time, so quite a while. Then it will stay as a child for another three hours before being useable.