Above is sound advice Xenon. It's a fairly simple process. Both my children were born in the Philippines and did not apply for their passports until they were 5/6 years. But even then it was not difficult. I just followed members advice on this forum.
One thing to be aware of is... The day your child gets their British Passport they will become British and subject to Immigration Laws in the Philippines just like you. Overstaying etc:
Sheldon