For Carol, with love.
I learned this from the singing and playing of Suffolk fiddler Harkie Nesling, on the Topic LP The Earl Soham Slog.
The song was also included on the Veteran CD Good Hearted Fellows. Mike Yates notes
The term ‘Teddy Bear’ was first coined sometime around November, 1902, when American President Theodore ‘Teddy’ Roosevelt was hunting in Mississippi. He had failed to shoot
anything, so friends captured a bear, which they tethered to a tree, and invited him to shoot it. Roosevelt’s reply: ‘Spare the bear. I will not shoot a tethered animal,’ soon became common knowledge and later that month Clifford & Rose Michtom of Brooklyn produced a soft bear which they called‘Teddy’. I would suspect that Harkie Nesling’s tune and short text probably date from the period 1902 up to the outbreak of the Great War in 1914, a time when Teddy Bears were very much in vogue and millions were sold in Europe and America. At least one other similar piece can be dated to 1907: this is Be My Little Teddy Bear by Vincent Bryan (best known for writing In the Sweet Bye and Bye) and Max Hoffman. Sadly, though, this is not the song that Harkie sings.
(from http://www.veteran.co.uk/vt154cd_words.htm#Teddy Bear)
You can hear Keith Summers’ recording of Harkie Nesling singing this song on the British Library website.
You can hear a 1907 recording of that other Teddy Bear song, sung by Ada Jones and Billy Murray, at http://www.youtube.com/watch?v=FHPeGd5qn9I

photo from http://pinterest.com/pin/63472675971011317/
Come and be my little teddy bear
Andy Turner: vocal, C/G anglo-concertina