Do I need an apostrophe after children in the following sentence?

We work on the product development on the children's and adults lists

Not necessarily, but either way you must mark "adult" as genitive: "adults'"

With NP-coordination the genitive case may be marked on each coordinate or just on the final one:

We work on the product development of the children's and adults' lists.
We work on the product development of the childrens and adults' lists.

It's a free choice: both methods of punctuation are perfectly correct.

PaulM