I would recommend point and shoot for a newborn, forget the brand, forget the features, they are "all the same".
What you need is usability (battery life and connectivity). Yes, you'll cherish these pictures forever, but they will all be static, close-up and relatively easy to capture. However, you'll want to easily get the images onto a phone/computer/web for the grandparents. Get one with wifi (and bluetooth).
Also choose something with proven integration - do you have an iPhone? No, seriously, my wife has been through 3 really good p&s, but now she only uses the iPhone ("too much hassle"). The kids don't use theirs because the battery is out, or it's a hassle using a card reader.
If you want a dedicated camera, great, check the wifi app, check the reviews for the app, some are less than flattering. Consider other features - are you in the southern, warmer states? How about a waterproof camera (with wifi) for beach/swimming/durability - there are some photographic compromises. Make sure it is drop proof (the camera, not the baby).
Step 2:
Having "gotten the bug", these kids realise that "the other baby" is on the flip-side of the camera, they want to see the image before it is taken!
Time to buy an SLR with faster focussing, startup and minimal shutter lag - do this before they are 2-years old or you will miss priceless moments (OK "mummy" won't/didn't, but I'm trying to capture this at the same time, and failed!)