Although it has a frivolous sounding title, this song is actually quite a sophisticated piece of classical music written in 6/8 time; it also has a rather complex history. The background to the whole teddy bear thing is outlined by Michèle Brown in The Little History Of The Teddy Bear.
In 1834, Robert Southey wrote Goldilocks And The Three Bears; in 1894 a German toy company came up with a stuffed bear; in 1899, Margarete Steiff registered patents for twenty-three of her soft toy designs including a dancing bear; in November 1902, Morris Michtom sold the first Teddy Bear in his Brooklyn shop. The year 1906 saw the first advertisement for the "teddy bear", in the trade journal Playthings, and in 1907, a book called Teddy Bear was published; written by Alice Scott, illustrated by Sybil Scott Paley, and The Roosevelt Bears newspaper strip was published in book form.
There are numerous editions of the music. The National Library of Australia holds a copy in its on-line digital collection, words copyright 1939 by Allan & Co of Melbourne. What appears to be a copy of the original is held by the British Library; this is "The Teddy Bears Picnic" [no apostrophe], copyright M Witmark & Sons, stamped received by the British Museum, August 11, 1909; it was arranged for Brass Band by Warwick Williams. This edition contains an amusing note to the effect that: "A very good imitation of a bear's growling may be produced by fixing a slack Bassoon reed through a piece of cork cut to fit the mouthpiece of an E[flat] Bass, or a Euphonium."
Another British Library holding, "The Teddy Bear's Picnic" [single apostrophe] arranged for Banjo by Robert Mahood, copyright 1908 by M Witmark, says this arrangement is copyrighted 1923.
This is all well and good, but the definitive version has lyrics by Dublin University graduate Jimmy Kennedy. In an interview published in the first issue of The Songwriter, dated June 1937, Kennedy said the music was twenty years old when he put words to it. This date may not be quite accurate, but the Kennedy version was first recorded in 1932 by Henry Hall and his Orchestra with vocalist Val Rosing.
In 1952, Kennedy published a children's poem The Story of the Teddy Bears' Picnic, credited by the Author of the famous Song "The Teddy Bears' Picnic". He appears to have been living in the United States at that time, because the self-published booklet was available from James Kennedy, 375 Park Avenue, New York at 1s6d or 20c; the same year, a coloured, fully illustrated version was published by the music publisher, B. Feldman of London.
Another arrangement of the Kennedy version, by Andrew Carter, in the Oxford Choral Songs series credits it "Original words and melody [copyright] 1907 and 1947 by Warner Bros...": This arrangement copyright 1989, published by Oxford University Press, Music Department.