In Barefoot Gen, Kimie goes into labor the day the atomic bomb falls on Hiroshima, after watching her husband and two of her children burn to death.
In Coppelion, Ibuki enters into labor when the ordinary humans of the cast desperately need to evacuate. The team attempted to avert this by getting her airlifted to a hospital, but couldn't get a helicopter. Just to make things even more complicated, they can't transfer her out of the sealed trailer because the area is irradiated and she can't get into a hazmat suit while in labor.
Inverted: In Chapter 12 of the Superjail! fanfic Extended Stay, the Warden and the Mistress get married while Mistress is 8 months pregnant. But just as they are about to kiss and wrap up the ceremony, her water breaks in front of everyone. A chapter later, she gives birth to mixed-gender twins.
Kyon Big Damn Hero example: Kyon was born during the Watanagashi festival ceremony. This was a heavy weight on his mother because, due to the birth's timing and the meaning of the festival, some of the Hinamizawa villagers believed Kyon was the representation of his mother's sin.
In Apocalypto, Jaguar Paw's wife Seven goes into labor while at the bottom of a pit that's filling with water from the heavy rain.
In Final Destination 2, the main characters believe they won't have to die if the pregnant woman gives birth, so the police picks her up and keeps her in a jail cell to make sure she is safe. Cue: her water breaking and the cop driving her to a hospital...Turns out she can't save them because in the initial timeline, while everyone died the pregnant lady actually survived.
In the Star Trek reboot film, James T. Kirk's mother goes into labour during Nero's attack on his father's ship. His father gets the news of his son's birth just before he gets killed.
In Terry Pratchett's Night Watch, Lady Sybil goes into labour just as Sam Vimes returns from a trip to the past. But Sam returns naked. He has to run accross the city in the nude to be with her. He also has to choose to be with her over and above hunting the villain who was returned with him. Sam decides Carcer the villain can wait.
Not really a "maternity" crisis, since in this case the birth is an egg hatching, but seen in the third book of the Temeraire series. Of all the times for a baby dragon to hatch, it would have to be just as the cast is on the run through enemy territory with almost no food, right?
Star Trek: Voyager: B'Elanna, the ship's chief engineer, goes into labor as the series finale is starting.
The trope has a sci-fi Up to Eleven moment in "Before and After". Kes is moving backward in her own timeline, and time-jumps into her earlier self just as she's giving birth on a shuttle that's about to rendezvous with a severely-damaged Voyager which is under attack by a hostile alien species.
Once on the most awesome moment of television ever (in-universe) on Robin's so-early-it's-still-late morning show.
Once at Marshall and Lily's wedding (the harpist).
Then Lily herself, when she's sent Marshall off for a relaxing night of drinking and gambling in Atlantic City to get him out of her hair.
On Bones, Brennan went into labor while investigating a homicide inside a prison.
In Highlander, Dr. Lindsey goes into labor after being caught in a tunnel collapse. Duncan ends up helping her deliver.
Played straight and averted by Good Luck Charlie. In season 1, the flashback to Charlie's birth has Amy fairly casually get up and get herself to the hospital. In season 3, for the 5th baby, Amy is a week late, and wants the baby out. She tries to induce pregnancy the day before Charlie's birthday, so the two youngest children didn't have to share a birthday. Naturally, she goes into labor the next day. Lampshaded at the beginning of the episdoe: "Wouldn't it be horrible if the baby came tomorrow?"
Eastenders has an episode where a woman goes into labor on a tube train that's being held at a signal.
Farscape has Aeryn going into labor in the middle of a war zone. She doesn't stop shooting her enemies for it either.
On Rookie Blue a pregnant cop goes into labor shortly after another female cop is kidnapped and the entire precinct is mobilized to look for her. The pregnant cop gets to the hospital without much hassle but the baby's father is in charge of the search so she forbids anyone to call him so he will not be distracted.
Sheppard: It's just that every time one of these things happens in the movies, the pregnant woman goes into labor.
Teyla: I am still a long way from my due date.
Sheppard: Yeah, it's the same thing in the movies and then, wham!
It was ultimately subverted that time, but only because there was an EVEN WORSE situation later (Teyla gives birth while trapped on an enemy ship.)
Happens during the Grand Finale of The New Normal with the surrogate mother Goldie going into labor with Bryan and David's son during their wedding.
The X-Files episode "Existence" is the exception mentioned above—the entire episode revolves around Scully leaving town to give birth, hiding from the Super Soldiers who want to kidnap the baby. However, while in hiding, her water breaks right after "killing" Billy Myles and having the rest of the Super Soldiers descend on the abandoned town.
The NCISChristmas Episode "Newborn King" had the team defending a very pregnant Marine from mercenaries trying to get the baby, due to it being, if male, the only surviving heir to an Afghan tribe (she'd fallen in love with the previous heir, who was later killed by the Taliban). To make matters worse, there's a power outage that knocks out the cell towers. Gibbs ends up delivering the baby in a gas station while Ziva fights off the mercs in the next room.
British series Call the Midwife, being about midwifery, features its share of such situations. Notably:
A woman falls while hanging the laundry. She has a concussion and is in labor, 9 weeks early, with her 25th child...and the only people home with her are children too young to help.
A Jamaican woman who can't get through to the midwives by phone walks all the way to the steps of Nonnatus House, then gives birth in the hallway.
The daughter of a Swedish sea captain lives in a cabin so tiny that there's no room to spread her legs to give birth. The midwives have to take a rowboat and then climb a rope ladder to get to her.
A woman goes into labor as her father is dying and refuses to leave his side, or lie down on a bed. She gives birth kneeling on the floor while Nurse Leigh lies on the floor to catch the baby.
On My Name Is Earl, Joy goes into labor with the baby she's surrogating for Liberty and Ray-Ray while Earl's old friend Frank holds her hostage in her own formerly Frank's trailer and makes her move the stove to get the money he hid behind it several years ago.
Pemma in The Legend of Korra is pregnant all through the first season with her and Tenzin's fourth child, and yet it waits to be born when the Equalists launch their full attack on the city and Air Temple Island.
In an episode of Family Guy, Peter gets into an altercation with a scruffy-looking individual, and punches that person in the face. Everyone is horrified and gives Peter a What the Hell, Hero? for hitting a woman, although Peter doesn't believe them. It turns out that the person Peter punched really was a woman, and pregnant, because she went into labor right afterward.
In Hey Arnold!, Arnold finds in his father's diary that when his father and one friend were taking his mother to the village and suddenly get caught in a volcano eruption, Stella went into labor. Luckly they found shelter in an ancient ruins.
This actually would have been a plot point in the unmadeJungle movie: apparently the eruption stopped right after Arnold was born, so the natives his parents were staying with believe him to be some sort of godlike figure.
A variant: Arnold and his friends are stuck in a stalled subway car with a few other passengers, including a Blind Black Guy whose seeing-eye dog suddenly seems very sick. It turns out to be a case where Your Tomcat Is Pregnant.