Hi Dean Welcome to the Forum
..yes i think you can register the birth of your child with the Philippine Embassy in London because i assume your wife Lucy is still a filipino citizen at the time of the birth of Lucita (baby) ??
please read:
http://philembassy-uk.org/faq/can-i-...ippine-embassy
...I dont think there is a need for PH passport for baby Lucita , if she got the British passport that is sufficient enough to go in and out of the Philippines.