Kids can be signed up to some kind of talent agency type thing. But babies are babies for not that longish, even if those used for just-given-birth scenes are much older than they would be.
One thing i’ve noticed is that often in commentaries or whatever one of the cast or crew will say “Oh, that’s my son/nephew/baby of a friend of mine” - how common is it just to get the use of a baby of someone in the production?
There are modeling agencies for babies, believe it or not. I had a class with a gal who did that with her infant son, it was an easy way for her to set him up a college fund.
As for using the baby of a relative, I’d imagine that’s somewhat common. Hollywood tends to have a lot of nepotism, and if the director, producer, or whomever, happens to know the parents, then they’d have some idea of what it’d be like to work with them, and wouldn’t necessarily have to deal with a “celebrity mom” type person.
Aren’t twins used wherever possible, to get around regulations and/or laws about how long a single baby can be kept on a set?
Interestingly enough, while you probably knew there are tons of rules about using children and babies, you might not know that there are rules about the kinds of jelly used to simulate birth goo on an infant. You can’t use strawberry and, I forget, maybe some other kinds, because of the allergen risk. Cherry is a common red goo that’s “safe”.
And, of course, when a mommy film and a daddy film really love each other . . .
Nowadays, newborns are often played by special effects – a doll or CGI. Obviously there are major advantages to using them (no regulations to follow, no child getting sick or crying on the set, no problems with retakes, etc.).
If you look at older movies, you’ll notice that the “newborns” were particularly well developed – they were large, could hold their heads up themselves, etc. Since it was hard to get a real newborn, movies would get babies who were a few months old.
God damn it. 
This one I actually know. When my wife was in the hospital when my first daughter was born, her roommate was a commercial actress. Our daughter was so striking that she encouraged my wife to bring her to auditions. Here’s the dope.
First of all, all (or most) babies are cute. Looks have very little to do with anything. The thing directors and casting agents evaluate is separation anxiety. They will be filming the kid away from its mother or father, and if the kid will start crying or reaching out, it is going to be very expensive. I’m not surprised by the nepotism - since all babies are equal, why not go with one you know?
My daughter didn’t separate all that well, and didn’t get an acting job until she tried again when she was 10.
It is true that newborns are played by older kids - there are either laws or union rules or both about this. And yes twins are nice, but rare enough so that it’s a problem not being a twin.
One warning. Many “talent agencies” for babies are scams. The scam is that they direct you to an expensive photographer to take head shots. The “agency” gets a cut. In fact, babies don’t need head shots, since they change appearance so quickly. My daughter’s manager used to be a child actor, and he was very proactive on warning people about this.
We went undercover to one such place in NJ, later shut down. They gave their pitch, and then separated the parents and kids, one to a room, clearly so that each and every one could be told how outstanding their kid was. (Our daughter was the oldest there by far, and already professionally represented, so we hoped they’d tell us she was no good. 
) NJ law requires that you tell people what percentage of clients get jobs. They showed us a contract, very quickly, then yanked it away. Then they made the mistake of leaving and we found another in the drawer of the guy’s desk. The odds are overwhelming you’ll never get a job, of course. They never actually told us the odds were there, and we would have never seen it if we hadn’t pilfered a contract form.
What they do, and all they actually say they do, is to send pictures to agents. Agents do look at every picture that comes in, so you can send a picture yourself with better odds, since the agents know who the scammers are.
Now one of the kids on the Cosby show actually got a job through them - and then split as soon as her parents figured out what was going on. Our manager represented her, in fact, and he was pretty amused when we told him that she was being used as the success story.
Sorry for going on so long, but I feel it is my duty to warn people about this.
I used to work in a building that had a child talent agency on the floor above us. Every couple of weeks or so a casting call would go out and I’d get to share the elevator with a thousand MILFs pushing strollers. It was kinda weird.
One of the extras on the Shoot 'Em Up dvd talked about this. The mother said she was approached by the (IIRC) casting director who heard that she was due to deliver twin boys just about the time they would be in principal photography and asked if they could use the two infants. The rule is something like no more than 10 minutes under the lights at a time, so twins are preferred. Of course, they had an anamatronic “stunt baby” as well as using some CGI.
Over teh course of a particularly big film’s production there is usually at least one or two members of the crew, or their partner, who will be due to give birth during the shooting schedule. I expect they take advantage of that likelihood where they can.
Your story about the “talent agencies” is depressingly familiar to stories I’ve heard about scam literary agents.
My kid went the nepotism route. I’ve been used in several ads that my agency has produced when the fees weren’t high enough to justify hiring a real model, and this particular client was satisfied with that. When a recent new catalog required family shots, they asked if I’d be willing to use my son, since that would eliminate separation problems and make everything go more smoothly (which it did).
He got paid the full amount, and a pro model was hired to be my “wife”. It shows in the final shots: she’d pose perfectly and look straight into the camera, while I’d be trying to get my son to smile and then only remember to look at the camera at the last second.