I'd suggest including the request with the birth announcement to be honest. Something like "<name> has arrived safe and well (insert vital stats if desired) - Mum and bub are doing fine however we would like to spend our first couple of weeks getting to know each other as a family. We will be happy to welcome visitors by arrangement after this time"

To be honest, I'm kind of planning something similar both for the first day after birth and then once I'm home from hospital depending on how I'm feeling, as I had a similar problem when DD was born.