After your baby is born, the hospital staff will advise you to wait for a couple of weeks (to allow transfer of papers from hospital to the registry) before you go to register your child's birth at the birth registry. The one on the island side is in Admiralty.
You can register your child's birth anytime at the British Consulate in Admiralty without doing it in the UK. See here:-http://www.fco.gov.uk/servlet/Front?pagename=OpenMarket/Xcelerate/ShowPage&c=Page&cid=1065721857576