The universe established in Children of Men is so dense it's hard to know where to begin when coming up with ideas for how the it might be expanded. But one thing is for sure: no matter what story you tell, it's probably going to be depressing. Yes, The Matrix also dealt with the extermination of the human race, but at least it had badass robots and kung fu. In the Children of Men universe, no one can have babies anymore, no one knows why, and everything goes to hell in a handbasket.
The most obvious direction would be to go back to when humanity first started to notice the eerie trend toward worldwide infertility. This can be seen through the eyes of a couple who are trying to conceive. As their attempts fail they, and the rest of humanity, slowly fracture as it dawns on everyone that this isn't an isolated issue nor is it one that will go away on its own—and no one will discover a reason, let alone a solution, anytime soon.
The book Children of Men is based on can be mined for even more world-building material. It's filled with more social eccentricities and more factions vying for power than the film has time to touch upon, like how newborn animals are treated as human children by those who wish to experience parenthood. Another area of exploration from the book that isn't mentioned in the film is a social class known as the Omegas—spoiled brats who live life luxuriously and are generally hated by most but are considered untouchable because they're young, the final remnants of a time when mankind could propagate the species. There are rumors that some countries outside the UK that have fallen into social and political disarray sacrifice Omegas during fertility rituals. What's happening in America during all this? Or in Japan, a country that had declining population rates long before mass infertility struck? When Kee gives birth to her baby, the first child born on Earth in decades, are there other women who find out they're pregnant at the same time? What are their stories?
A Children of Men expanded universe wouldn't be the happiest series ever, but it could be among the most daring and poignant.