The story of Peter Pan in Kensington Gardens is so different from the well-known Disney tale that I was surprised how sad it was. Poor Peter - "betwixt and between", neither a bird nor a human - was forever one week old, having flown away from his home. He forgot he was a human and thought he was a bird until he landed on the island in the Serpentine. There, Solomon Caw gently explained that he had no feathers, nor claws, and was wearing a nightgown, so could not be a bird. He decided sadly that he should return to his home, but found he could no longer fly.
After some adventures he did get to his home, and he could have stayed with his mother, but didn't want to miss out on the fun he was having outside. He said he would come back and did eventually, but the window was barred and he could see his mother inside, cuddling another child.
He had lost his chance.
Don't worry though - he was quite happy living in Kensington Gardens. (I wouldn't like to leave you miserable.)
This tale is from "The Little White Bird" and appears as just part of the "Peter Pan in Kensington Gardens" book. It has absolutely beautiful illustrations by Arthur Rackham (above, 1906) and Mabel Lucie Attwell also illustrated the story (c1924). To the right is Peter trying to get in the barred window (don't look for it in my shop pages because I've sold that one, but there are other Attwells there).
If you want a (slightly) more Disney-like Peter Pan, you'd need to look at Barrie's "Peter Pan and Wendy".