Both sons born in Cyprus - #1 at BMH Dhekelia, #2 at TPMH, whilst I was serving at Nicosia just after the Turkish Invasion. Both were issued Birth Certificates under the special provisions Act 0f 1957 - should have a Form ORS 7. Applied for passports for both as soon as 16 - son #1 had no problems as Nicosia was still RAF, and BC clearly stated British Military Hospital and RAF Son#2 had more problems as BC just said TPMH and by then we were UNPA Nicosia, but eventually granted and has had no problems since. It seemed to be luck of the jobsworth at the time.
IIRC at the time there were lots of problems with very large Turkish women trying to get through the fence into SBA North so as to be rushed into BMH and thus claim British Nationality for child!