I think you may be able to qualify for a British citizen passport yourself. But that is based on the chain of descent from grandfather to grandson being purely legitimate (within the bounds of matrimony).
Also be aware that that right is likely not transmittable to any of your children born abroad. Also be aware that I am not a lawyer or immigration advisor and that you may want to seek professional advice.
As your grandfather was mentioned on the certificate of naturalisation of your great-grandfather in the UK, he was naturalised in his own right in the UK as well.
Section 27(2) of the British Nationality and Status of Aliens Act 1914 wrote:Where in pursuance of this Act the name of a child is included in a certificate of naturalization granted to his parent, or where, in pursuance of any Act repealed by this Act, any child has been deemed to be a naturalized British subject by reason of residence with his parent, such child shall, for the purposes of this Act, be deemed to be a person to whom a certificate of naturalization has been granted.
Your father was born a British subject by birth in St Lucia in 1938. He became a CUKC in 1949, in accordance with
Section 12(1)(a) of the British Nationality Act 1948.
As your father was a CUKC by birth on St Lucia, you would have become a CUKC by descent when you were born in an independent Trinidad and Tobago in 1965, under
Section 5 of the BNA 1948.
The Immigration Act 1971 also added a requirement for a person to hold the
Right of Abode in the UK. Your father qualified for the Right of Abode under Section 2(1)(b)(i) of the Immigration Act 1971, linked to above and you should qualify under Section 2(1)(b)(ii).
Because you held both a CUKC by descent and Right of Abode on 31st December 1982, you would have become a British citizen on 1st January 1983.
To make your passport application, you will need a copy of your great-grandfather's naturalisation certificate, which includes your grandfather's name on it , and the chain of birth and marriage certificates (keep in mind that the above only holds true if all the people involves were born while their parents were married), so the marriage certificates for your grandparents and your parents and the birth certificates for your father and yourself.
You will also likely have to write out a cover-letter explaining all the laws I mentioned above as it is unlikely that the caseworker will know how to deal with an application from the grandson of a person naturalised more than a 100 years ago, with its myriad changes of the law.
Having said all this, let me also warn you that moving to the UK at your age may provide some challenges that you should plan and prepare for.
As I mentioned above, your British citizenship won't be transmitted to any children born to you outside the UK and if you do have any children over the age of 18, you can't sponsor them to the UK on any family visa. You will be able to sponsor your spouse and any children under 18, but the visa application fees are quite high (~£3300 per person) and have a high minimum income requirement (from £18,600 per annum onwards).
You would likely also not be eligible for many benefits as many of them have a minimum residence period to qualify. The minimum residence requirement for
PIP (disability benefit), for instance, is two years.
As you have not worked in the UK, you will also not qualify for a full
state pension at the age of 67 - the state pension age in the UK (unless Trinidad and Tobago have an agreement in place with the UK on benefits and pensions), which is based on the number of years that you have worked and earned in the UK.
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.