I have found White Coral Bells, under the title of May Bells, in "Fourth Year Music," edited by Hollis Dann, published by the American Book Company, of New York, Cincinnati and Chicago, copyright 1916. As presented in this collection, White Coral Bells is exactly the same as I heard it in the 50's, both words and tune, except for the title. Since no composer credits are given, although they are provided for the majority of songs in the book, the song might easily be much older than 1916.

If the flower "white coral bells" is different from "lilies of the valley," as described above by "Q" on 5/10/07, then it's easy to see why the title of "May Bells" would have been more appropriate than "White Coral Bells;" the latter would only relate to half (1) of the flowers mentioned in the song. On the other hand, the explanations presented above, that "white coral bells" were part of a device made from white coral in colonial times, also seems plausible.