This plot sounds so familiar....I think I must've read it in passing somewhere, I feel like for an episode of an anthology show like Outer Limits or something.
Anyways, this film, Delivering Milo (2001), seems to match:
Elizabeth believes she has gone into labor, but at the hospital, the
doctor tells her she was having Braxton Hicks contractions. The
reality is that in the world from which babies come, Milo is too
scared to leave the comfortable place where he has lived. Since babies
must be born in order, no other babies will be born until Milo changes
his mind. The people in charge give the job of educating Milo to
Elmore, who is in a kind of limbo until he proves he is worthy of
Elmore and Milo go out the door which normally serves only as an
entrance. Outside is the energetic and terrifying New York City.
Elmore loves it, but Milo doesn't want to be in this scary place.
The big news story is the lack of births anywhere.
Elmore eventually takes Milo to Atlantic City, where he loved to
gamble when he was alive. Elizabeth also ends up in Atlantic City.
Milo's mind is not changed and there is a deadline. A door will close
and no more babies will ever be born. In Atlantic City, Elizabeth
meets Milo, who realizes Elizabeth is his mother, and decides he wants
to be born. He also realizes Elmore is the father who abandoned
Does that sound like the work you saw?