This isn't quite what you're looking for, but you might be interested in the Gynecology of Soranus of Ephesus. He was a Greek doctor active in the beginning of the 2nd century AD. The Gynecology is a scientific/medical text, not really a late antique "What to Expect When You're Expecting"-type guide, but it contains passages on things like:

• how to swaddle a newborn

• how hard a surface to place it on (not too hard, which will dent the soft back of the skull; but also not too soft, which can prevent the backbone and muscles from strengthening)

• advises on selection of nurses--when the mother should nurse her own child, what to look for in a wet nurse, etc

• teething, learning to walk, dealing with childhood illnesses

The table of contents in English is available online here, but unfortunately you'll need access to ACLS (probably through a university) to read the actual text.