“When I was growing up, I watched ‘The Smurfs’ on Saturday morning,” said Rich Greenfield, an analyst specializing in technology and media at BTIG Research, referring to the NBC cartoon show from the 1980s. “That was the definition of kids’ TV. Kids today don’t even think about turning on the TV networks or cable on Saturday. They want it on demand, and that’s how they’re used to getting it.”

For the streaming services, in particular, their treasure trove of data clearly show what their viewers crave: According to Netflix, more than half of its 93 million members in nearly 200 countries, for example, regularly watch its children’s shows.

“Children’s content is important, because it’s our first opportunity to build brand love for Netflix that we hope will last a lifetime,” said Andy Yeatman, director of global kids content at Netflix.

The real targets, of course, are parents who pay the subscription fees. “Kids’ content gives the household a reason to subscribe, because kids watch every single day,” Mr. Greenfield said.

Following their schmooze with the puppets, Ms. Andrews and Ms. Hamilton discussed the birthing process for their show over scones and coffee in a conference room at the Creature Shop. The women have an easy rapport, built up not only as mother and daughter but also as the co-authors of more than 30 popular children’s books, including the “Dumpy the Dump Truck” and “The Very Fairy Princess” series. (Ms. Andrews has four other children, including two stepchildren, as well as 10 grandchildren and three great-grandchildren.)