When my sister was expecting, we had a family meeting as to who would be told what, when, and how. She had a list of who she wanted at the hospital (five people), who could be told same-day (five more people, by text message or phone call only), and everyone else could wait until she was home and settled before they knew the details.
It ended up being that she didn't wait long enough, as my niece wasn't very well at first and had to spent nearly all day every day on a bili-bed. Sis wasn't sleeping well, the kid wasn't sleeping well, and as soon as the announcement went out, sis got innundated with phone calls and text messages and people stopping by. They'd want to see the baby, who really did need to be left alone on the bili-bed, or the phone would wake the baby or my sister and she wasn't getting any real rest at all. My mom ended up sending out an email that mother and baby were doing as well as could be expected, but to PLEASE leave them alone until further notice. She got called rude. She took my sister's phone away and screened calls, lived in with her to help take care of the baby and be the mean one to turn people away, and my sister was miserable.
If this family wants to wait a week to announce the birth, that's their decision. I'm firmly supportive of them. And if they later decide that a week isn't enough time, then that's fine too. It's their child, it's their lives being disrupted, they have every right to spread the word when and how they feel comfortable.