As might be deduced from its title, "Nellie The Elephant" is a novelty song, the main novelty being in the claim that she packed her trunk and said goodbye to the circus. Presumably she left in a taxi after making a trunk call to a local cab firm.
The song was written by Ralph Butler and Peter Hart. According to Butler's obituary in the London Times
of April 9, 1969, it was a song that was not entirely out of character. He is said to have won an Ivor Novello award for it at the age of 70, and to have co-written both "Hey Little Hen" and "Run Rabbit Run". The award - Outstanding Novelty Song - was the same as that won by The Singing Postman ten years later (Best Novelty Song) with "Hev Yew Gotta Loight, Boy?