"what hair colour have you got?" or "what colour hair have you got?"
Leaving aside have you got/do you have: Let's look at how we talk about characteristics or descriptions (color, size, length, height, weight, etc.) of things in English:
Normally, these would be questions that are: What + characteristic + noun:
What color uniform have you got? What color hair have you got? What length surfboard have you got? **What size house have you got?
In that sense, /What color hair have you got/ is a standard pattern (structure) in English** (any variety) for asking question re some characteristic of a thing.
This standard pattern may be contrasted with questions formed with "quasi"compound nouns made up of NOUN + NOUN which are formed from NOUN OF NOUN: - hair color=color of hair - car model= model of car - surfboard length=length of surfboard - dog type=type of dog
All of the above may also be made into a question using WHAT:
- What hair color have you got?
- What car model have you got?
- What surfboard length have you got?
- What dog type have you got?
So, there are two different grammatical structures in play, both of which are standard and normal in everyday English.
Please note: Have got/have is another issue but regardless: /Have you got a car/ is semantically equivalent to /Do you have a car/, and English has two ways of saying exactly the same thing. And, sometimes, it's confusing for students to realize that as when you get into the present perfect, there is one huge difference here between BrE and AmE but that is not relevant here in the present tense.
"What colour hair have you got?" is certainly idiomatic in British English, meaning "what colour is the hair on your head?".
"What hair colour have you got?" would be understood, but does sound slightly weird. It would make more sense when talking about hair dye ("hair colour" as a compound noun can be used to mean hair dye).
I'd post an ngram as evidence, but you already posted it in your question. So my only source is myself: native BrE speaker, south-east England.