Yes, your baby is British by decent.

http://www.ukba.homeoffice.gov.uk/br.../bornoverseas/

British citizenship may descend to one generation born abroad. So if you were born outside the United Kingdom or qualifying territory and one of your parents was a British citizen otherwise than by descent, you are a British citizen by descent.

It is your choice if you register the birth at the embassy.

http://ukinthephilippines.fco.gov.uk...h-registration

There is no requirement for a consular birth registration to be done for any birth that has occurred overseas. The original birth certificate issued by the authorities in the country in which the birth took place, along with a notarised translation if necessary, is sufficient for all purposes in the UK (including passport applications). The advantages of a Consular birth registration are:

  • A British style birth certificate is available as evidence of British nationality
  • A record of the birth will afterwards be held in the General Registry Office in the United Kingdom