OK, yeah, so I'm feeling a little silly today. This is an old one and undeniably goofy, but I guarantee it will be marching around inside your head for hours, if not days, now.
The Toy Dolls are a punk band out of Sunderland, England, originally formed in 1979 and are, surprisingly, still at it (after a gazillion personnel changes - 14 different drummers and 12 different bassists). Nellie the Elephant, by far their most well-known song, was written in 1956 by Ralph Butler and Peter Hart.
Nellie the Elephant by The Toy Dolls (released as a single in 1982) from the 1983 album "Dig That Groove Baby"
The Toy Dolls website