I had to get in touch with some records office in the UK as he didn't have his birth certificate, took a couple of weeks.
You need to contact the Registrar of Births, Marriages and Deaths for the area in which your father was born. This can be a little difficult to track down if your UK geography isn't that good, and is further complicated by the fact that it's done at a different level of local government in different areas of the UK (for instance in England it's done at City/District level in urban areas and County level in rural areas, I'm not sure about Wales, Scotland or NI).
Another thing that may or may not help is the fact that the Republic of Ireland was a part of the UK prior to 1922 and therefore descendants of people born there have as much of a claim to British citizenship as descendents of those born in NI/Mainland Britain.