That came from this site and a thread asking what songs Mums sing to their children these days.

http://www.netmums.com/coffeehouse/general-coffeehouse-chat-514/coffee-lounge-18/749712-songs-you-sing-your-little-ones-your-parents-family-used-sing-you.html

The use of the term teddy bear dates the song to be after 1902 but the reference to soldiers and sailor boys possibly makes it post World War 1
song. I think the maestro searcher will probably come up with more soon.