In Cyprus, no problem if the child was born in the military hospital (Akrotiri in my son's case). Otherwise same applies as above.
Akrotiri is a unique case. The Hospital is in the Sovereign Base Area, it is not in the Republic of Cyprus.
People born there have trouble today getting USA visa's. Because UK passport says place of birth Cyprus.
Two exceptional cases that I am aware of.
I was stationed with the RAF Liaison Party at Istres, France. When wives gave birth in local civvy hospital, husband drove rapidly to Marseilles and the British Consulate to do necessary paperwork, within 48 hours of birth.
A friend of mine; his father was Irish, in 1922 he took the option to stay British.
He traveled a lot, and had a son born in Argentina. This son became my friend.
Son joins RAF as a Boy Entrant, while serving he gets married, wife has 2 children, one born in Malta one in Germany.
In later civilian life, my friend decides to emigrate to Australia!
Oz authorities demand to see copy of document signed by now deceased father in 1922!
Somerset House produced it!!!