British nationality law is incredibly complex, but the general picture is as follows:
After the Second World War, the formerly unified British Subject status that applied throughout the Empire (later Commonwealth) started to get subdivided into national citizenships. Canada, Australia, New Zealand, South Africa and other self-governing countries within the Empire began to enact their own citizenship laws. So, e.g., an Australian would become both a British Subject and a Citizen of Australia. The residual category, for British Subjects who didn’t acquire Australian, Canadian, etc citizenship was “Citizen of the United Kingdom and Colonies”, often abbreviate to CUKC. The OP, both by virtue of birth in the Bahamas, which was then a colony, and by virtue of being born to a CUKC father, would have been born a CUKC.
As colonies transitioned to independence, each enacted its own citizenship law. There’s a link upthread to the Bahamian citizenship law. It’s fairly typical, in that anybody who was a CUKC by virtue of birth in the Bahamas, or birth to a father born in the Bahamas, was granted Bahamian citizenship. And the UK’s legislation granting independence to the Bahamas, also linked upthread, is again fairly typical; a CUKC who acquired Bahamian citizenship lost their CUKC status unless they had some independent claim to that status, not relying on their connection with the Bahamas.
There were further reforms in 1981, as part of which the CUKC status became “British Citizenship” and British Subject status, formerly an overarching category embracing all Commonwealth citizens, became a residual category for people who have a connection to the UK but don’t hold citizenship in any Commonwealth country (including the UK itself).
It’s likely that the OP is a Bahamian Citizen from birth. It’s not likely, unless there are additional links with the UK that have not been mentioned, that the OP is, or can become, a British Citizen. if, contrary to my expectation, the OP has lost Bahamian citizenship somewhere along the way, it’s possible that they could claim British Subject status. But as British Subject status doesn’t actually come with a right to enter or settle in the UK, it’s not hugely useful.
As a Bahamian citizen the OP enjoys visa-free entry to the UK (and indeed the whole EU) but not the right to settle there. However if by any chance any of their grandparents were born in the UK then, as a Bahamian citizen, the OP could apply for an ancestry visa entitling them to live and work in the UK for five years, with the possibility of extending that time. And, after living in the UK for five years or more, the OP could apply to be naturalised as a British Citizen.