The moment after a baby is born, you can click on it and select "age up." Just like that...
For children and for sims of any other age, any of the birthday cakes can be used to make them age up in the way that they were used in the Sims 3 (I believe at any time as well).
UPDATE: Sims do in fact age up automatically.
CORRECTION: There is a way to disable aging in the options menu. Below, I had incorrectly stated that there was not.
However, I am not sure if the birthday cake is actually the ONLY way to make them age up. If so, that would mean you could keep your sim at one age, or state of development, indefinitely, which actually might be nice since the "disable aging" feature from the Sims 3 is now gone. I'll update my response if I find out whether or not the sims age up automatically at any point.