"Bob's your uncle" ... no he's not!

As far as I know this is only heard in British English. I've never heard Americans use it unless they were imitating Britons. Etymology here:

Bob's your uncle - ironic expression of something easily done - like: there you have it, as if by magic - Cassells cites AJ Langguth's work Saki of 1981 in suggesting that the expression arose after Conservative Prime Minister Robert (Bob) Cecil appointed his nephew Arthur Balfour as Chief Secretary for Ireland in 1900, which was apparently surprising and unpopular. In this sense the expression also carried a hint of sarcastic envy or resentment, rather like it's who you know not what you know that gets results, or 'easy when you know how'. Since then the meaning has become acknowledging, announcing or explaining a result or outcome that is achieved more easily than might be imagined.

Bob's your uncle, an expression meaning "everything will be fine", originated when Arthur Balfour was unexpectedly promoted to Chief Secretary for Ireland by the Prime Minister, Lord Salisbury, in 1900. Salisbury was Balfour's uncle and his first name was Robert.

Other notables named Robert have also been nominated as the origin of this phrase. See here for more.

Green's Slang Dictionary notes the gap between Balfour's appointment and the appearance of the saying and suggests that Bob in this case is a euphemism for "God".

Edit Michael Quinion has more on this (calling into question the Balfour theory) here.

Several respondents have stated that "Bob's your Uncle" is not American English. I have spent most of my life in North America and have regularly heard that phrase used in the sense of "before you can say 'Bob's your Uncle'" - indicating, as suggested, simple, easy, and taking little time. There was no suggestion that British English was being imitated or was in any way involved.

I have certainly heard it used in Australia. When I lived there in 1974/5 the safety of vasectomies was being questioned. I distinctly remember a tabloid newspaper headline which read 'One slip, and Bob's your Auntie', confirming to me that the Australian sense of humour owed much to the old country.