I was born in Switzerland 2 weeks before WW2; my birth was registered with the consul in Zurich but I never had a birth certificate. After the war I needed one and my Father and I went to Somerset House; my record was located in an Odds and Sods section which dealt with births and deaths abroad, those drowned at sea etc. That said I do not think that there has been anything at Somerset House dealing with these matters for years, it was all moved up to Southport SFAIK.
My daughter was born In Akrotiri and as it was British territory (SBA), she got some variant of a normal UK Birth certificate
Link to Southport Office
Copy of birth, death and marriages certificates