The expression, "Bob's your Uncle" is very common here in Oz, and the definition of fanny is the same as in the UK. That definition colours all sorts of phrases and situations in Oz which would be fairly neutral in the U.S., e.g. Sweet F.A, or Sweet Fanny Adams.

I had a friend, now deceased, who used to write a very clever, very funny one page article every month in a motorcycle magazine. One of his expressions, both in writing and verbally was, "Robert's your father's sibling". He often used it if he had just explained how to do a complex bit of mechanical work, etc.

This article
'Bob's your uncle' puts forward a reasoned suggestion that the phrase became popular after a music hall song in 1924:

Bob's your uncle
Follow your Uncle Bob
He knows what to do
He'll look after you

The author of the article proposes that the reference to Robert Cecil and his nephew, i.e. the incidence of nepotism, was retrofitted to the phrase after the phrase became popular.

I never liked that nepotism explanation for some reason. Possibly because the place where I have worked for over a decade is totally run on nepotism: "not what you know but who you know", so it gives me an icky feeling even hearing the word nepotism.

Helen