ur-text for the zombie, as people today think of them, is the 1968 film 'Night of the Living Dead'. Famously, the original film is in the public domain owing to some error made by the production company. So my question is, does that mean the story itself is public domain, too? For example, would it be legitimate to produce a book using the same name, character names and plot? Or, failing that, would it be legitimate to produce a book that copied the story but went under a slightly different name (e.g. 'Night of the Zombies'), and changed the character names? Copyright experts agree that the 1968 film, Night of the Living Dead, is in the public domain in the U.S. and -- thanks to a case involving a 1962 John Wayne film -- so is the underlying screenplay. That would leave others free to duplicate the story, plot and dialogue and create derivatives. The characters such as Ben, Barbra, Harry, Helen, Tom, and Judy, would also fall into the public domain, as this case has indicated. Finally, the title of the work has also passed to the public domain as the Supreme Court has held that the title of a work that enters the public domain cannot be protected as a trademark. Which means that a children's re-telling, as you propose, would not violate U.S. copyright law. However, your derivative work could only be protected to the extent of any new or "incremental" material added to the original.
Muddy waters. Although the story, title, plot, dialogue, and characters are in the public domain, the original screenwriters (George Romero and John Russo) went gone on to create copyrighted sequels and remakes of the original using the same characters. (The two screenwriters couldn't agree on a sequel and Romero acquired the right to movie sequels using "Of the Dead" and Russo acquired rights to "Living Dead" movie sequels. Russo also went on to create a series of Night of the Living Dead comics using the characters and plot from the original.) No trademarks have been federally registered for "Night of the Living Dead" for books but the comics publisher has a reasonable claim that the term is used as the trademark for the series. All of this adds to the confusion surrounding copyright (and explains why a squadron of would-be and oft-bogus enforcers swoop down in opposition whenever the film is loaded on YouTube).
Bottom Line Dept. We think the coast is clear for a young reader's edition in the U.S. but you may want to get an opinion letter from a copyright/trademark lawyer, if possible. You have a good chance of prevailing on copyright claims but we think your activity may possibly attract lawsuits, anyway. In addition, we can't confirm copyright status of the film in the U.K. (where you're located).
BTW ... For those interested in "monetizing" the film, it's also unclear whether certain "separable" elements of the film are also public domain -- for example, the music (credited at different times to William Loose and Fred Steiner and to Scott Vladimir Licina), and the trailer for the film.