There are many ideas about the meaning of this nursery rhyme.
Here are two, but, personally, I think it is just a great bit of nonsense
and the meaning is in the readers head.

In England's royal court, it was the custom to dub people with silly
nicknames. Elizabeth I was often called The Cat from the way she
fiddled with her Cabinet ministers as if they were mice.
The cow, moon and her lap-dog were also characters in these court
charades. The dish was Elizabeth’s serving lady and the spoon was,
of course, the royal taster.
When these last two secretly eloped, Elizabeth had them captured
and confined to the Tower of London.
Nobody fiddled with Elizabeth!

Or is it Richard III's path to the throne of England:
Diddle diddle was the way he got rid of Edward V,
William Catesby was the cat and the pre-contract was
the fiddle. The cow was the Neville emblem, and the
Nevilles eclipsed the Percys, whose emblem was a moon.
Viscount Lovel was Richard's best friend, and his emblem was a dog.
The dish was Richard himself who ran away with the annointing
spoon at his coronation

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