Will babies eventually grow into a normal aged sim? If yes, how long will it take for it to grow old

It took my baby five days to be an adult but it depends in how much you look after it.
In order for them to be playable you need to have an unlocked and available sim slot, then tap the family portrait, click on your aged uo sim and press the promote button.

It will take less than a week, but when they grow up you won’t be able to control them

Yes. It normally takes 24 hours for wash stage of the child to grow up. Typically 4-5 days until they are fully grown