It's certainly a common expression. "I haven't a clue" is a slightly abbreviated form of "I have not got a clue". Said in response to a question you do not know the answer to, it literally means that, not only do you not know the answer but neither do you know anything which might act as a clue towards the correct answer.
It might well be considered an idiom, although strictly speaking, an expression is an "idiom" when its meaning as an expression cannot be determined by analysing the individual words alone. The meaning seems pretty straightforward to me.
I wouldn't really think of it as slang either for the same reason - it isn't replacement language - although as you point out, in some 'cockney' London accents (and probably others too) it is heavily abbreviated, dropping the "I" and simply saying "haven't a clue". That could fit the definition of slang as "words and phrases that are regarded as very informal, are more common in speech than writing, and are typically restricted to a particular context or group of people".
Shortening it that way might be typically British English, but as an expression it isn't restricted to the UK. US English speakers may favour the word "clueless" (the title of a 1995 US movie), or the similar "I don't have a clue" (the title of a book by US author Sarah Cortez), which again go to proving that "haven't a clue" isn't strictly an idiom because it's meaning is tied to its component words and can mean the same formatted differently.
