The first zombie children I remember seeing in a movie were in Resident Evil. I may be wrong, but it seems unlikely to me that it was the first zombie flick to include them. What film was the first to feature onscreen zombified children?
Examples drawn from this article:
The first seems to be Karen Cooper in Night of the Living Dead in 1968.
Dawn of the Dead featured Tom Savini's niece and nephew in 1978.
Burial Ground: The Nights of Terror had a zombie child (played by a 20-year-old midget) in 1981.
Redneck Zombies had a zombie toddler in 1987.
SPOILER: There's an earlier and (still quite powerful) example, from the great-grand-daddy of this genre, in "The Last Man on Earth (1964)". That's probably the earliest example of a film we'd recognize as a "zombie" movie today, in fact. So, child monsters were a part of this genre pretty much from the very beginning!
"Village of the Damned (1960)" and "Invaders From Mars (1953)" and other alien-invasion movies from the 1950s might technically have played around with the idea of kids being turned into soulless monsters. This sort of film is sort of a twin sister of the modern "zombie" movie, in which nightmarish and poorly-understood forces turn your neighbors into soulless monsters while your parents' comfortable, orderly civilization crumbles around you, and I think these alien invasion films could technically count as a forerunner of the "zombie" film, and be fairly considered as even earlier examples.
There might be some old-fashioned (Hollywood voodoo) zombie movies from the 1930s or 1940s in which kids get turned into zombies, but I'm really not familiar enough with that genre to point to an example. I don't think those older voodoo zombie films are thematically related to the sort of story you are talking about at all, though, so I couldn't really count them.
Zack Snyder's "Dawn of the Dead" remake in 2004 featured a kid zombie at the beginning. That movie really started the whole resurgence we have today. I can't rmember if there were dead kids in the original though. "Pet Sematary" in 1989 also had a child that came back alive again. Though, while he was crazy and deadly, he wasn't your typical "zombie".
They're the two I know off the top of my head. But I'm guessing that if you look closely at all of Romero's you would see a kid or two in the background. But they don't feature them.