It was probably during her childhood that Queen Elizabeth I enjoyed teasing her ministers. Some claimed that it was similar to the way a cat plays with mice. The Queen enjoyed dancing to fiddle music.
Moon and Lap-Dog were nicknames for two of her advisors.
Spoon was the nickname of a lady in waiting. Spoon was also a taster who sampled the Queen's food before she ate.
Dish was the nickname of the court who brought the Queen's food to her.
Dish and Spoon eloped. Yes, the Dish did run away with the Spoon. Later, someone created this rhyme.
These facts were found in an article about "Mother Goose" in The World Book Encyclopedia. There was no information about how the cow fit into the story.