The situation is not as straightforward as it seems.
Before 1983, your father was a Citizen of the UK & Colonies (CUKC). However, the British Nationality Act 1981 split the former CUKCs into three categories;
- British citizens - CUKCs who had a connection with the UK by "birth, adoption, naturalisation or (except as mentioned below) registration in the United Kingdom or in any of the (British) Islands;"
British Dependent Territories/Overseas Territories Citizens - CUKCs who had a connection with one of the British Dependent Territories (later called British Overseas Territories), but not with the UK.
British Overseas Citizens - Any CUKCs who did not fit into either of the two above.
St Kitts and Nevis was a BDTC in 1983 before independence. Therefore, unless your father had had the foresight to register as a CUKC in the UK, I believe that he did not become a British citizen, but a British Dependent Territories Citizen in 1983.
When St Kitts and Nevis became independent in 1983, British Dependent Territory citizenship was lost if there was no connection with any other British territory. British citizenship itself was not lost if your father had the foresight to register himself as British.
So, check with your father if he had registered as British before 1983. If not, I believe that he does not have a right to a British passport, either as a British citizen or as a BOTC.
Mind you, I am not a lawyer and my interpretation of the law could be wrong. However, you can see that the law is very complex in this field and that you may need legal advice as to the precise status of your father before proceeding.
Further reading:
Independence
British Nationality Act 1981, particularly
Section 11
Immigration Act 1971 (as enacted), particularly
Section 2
I am not a lawyer or immigration advisor. My statements/comments do not constitute legal advice. E&OE. Please do not PM me for advice.