Q:

Is the Ministry of Magic part of the British Government? 

It's not, by its own admission, a Government in its own right, but just a ministry:

(in certain countries) a government department headed by a minister. 
  "the Ministry of Defence"
    synonyms:   government department, department, bureau, agency, office
  "the ministry for foreign affairs"
  ~ Google

Did JKR intend to make the MoM part of the British Government?  Or is this a mistake of language? 
I do recognise that the Muggle PM in the later books does not recognise Scrimgeour, but I don't find this conclusive (Wizards could be classified above the PM, or just be on a Need to Know basis)

A:

No.  Absolutely not. 
I see no reason to presume that the Ministry of Magic is anything other than a wholly separate government, one that possesses its own Executive, Legislative and Judicial functions along with its own currency, international policy, etc etc. 
If I can quote from the pottermore article on the Ministry of Magic:

The Muggle Prime Minister has no part in appointing the Minister for
  Magic, whose election is a matter only for the magical community
  themselves.  All matters relating to the magical community in Britain
  are managed solely by the Minister for Magic, and he has sole
  jurisdiction over his Ministry.  Emergency visits to the Muggle Prime
  Minister by the Minister for Magic are announced by a portrait of
  Ulick Gamp (first Minister for Magic) that hangs in the Muggle Prime
  Minister's study in number 10 Downing Street. 

Since in the UK, all authority ultimately flows from the Queen, any organisation that doesn't fall within her jurisdiction is by definition not part of the UK government.