There is hope for HP fandom yet.
Ok you are one of the few fic writers that gets children correctly. Do you have any children? Cause it would make sence if you do. People seem to love making kids baby talk. Hell I have read fics where children at the age of 3 or 4 talk like one year olds or younger. The children you wrote were perfect for their age. Lily acted a little too young since she is supposed to be only 2 years younger than James which would make her 4 or 3. Either way, a kid that old would not act that young and would not need a "baby chair". Mostly after year 2 at the latest they don't use them anymore. Also by age 3 they talk in full sentences.
Btw, this is no harsh critisim, this is coming from a person that worked at a daycare for half a year. Lol. Everything else about your story and setting is either spot on or it is believable since it is a magic preschool. Lol. I am REALLY impressed with this fic. You have no idea how I avoid fics with kids in them because they get screwed up most of the time. Your fic makes me have hope for fic writers. Lol. Good job.