A few more suggestions:
"The Teddy Bear's Picnic," "The Happy Wanderer," "The Marvelous Toy," -- all at this website!
Rounds: "White Coral Bells," "Dona Nobis Pachem" (or "Yasem Lanu Shalom", if you prefer Hebrew), "Kookaburra," "The Snap, Crackle Pop (Rice Krispies) Song."
Nonsense songs: e.g., The Huma-huma-nuka-nuka appawaha Song about a small Hawaiian fish (I believe), Boom, Boom Diddum, Diddum, Mairzy Doats, which has a second verse about whales, eels, seals, fish, oysters and squid, ("A squiddley-doysters too...").
Movies: songs from Hans Christian Anderson ("The Ugly Duckling," "Inchworm," "Thumbelina," etc.), Mary Poppins, the older Disney cartoon features. What about songs besides the obvious O.T.R. from The Wizard of Oz? Check out pre-1970's musicals, and operettas like those of Gilbert & Sullivan to broaden young tastes before adolescence sets in. Rent the movie version of Pirates Of Penzance, starring Linda Ronstadt (the U.K., NOT the Aussie. version, which is too racy for kids) and see if any of the music appeals to your children.
Folk-Rock: James Taylor's "Sweet Baby James" and "You Can Close Your Eyes"; "Joy To The World" (a.k.a.: "Jeremiah Was A Bullfrog").
Albums: Peter, Paul, and MOMMY, any of Burl Ives recordings for children. Free To Be You and ME, mentioned by rechal, is an excellent P.C. collection; "I'm Being Swallowed by a Boa Constrictor", from this album, is great giggling fun to act out!
Book of Lyrics: Rise Up Singing, The Group Singing Songbook, by Peter Blood and Annie Patterson, has words and chords to 1200 songs, many of them suitable for children, in a deceptively small, child-friendly, paperback volume.
Some personal favourites: "Wynken, Blynken and Nod," "Cockles and Mussels," and "Mockingbird."
Hope this is some help.