Graham,
I stand to be corrected here, but as I understand the 'rules' there's no possibilitity to secure a UK birth certificate.
The best that can be done is to register the birth with the British Embassy in Manila or the FCO (Foreign & Commonwealth Office) in London. There's no legal requirement to do this at all, but it does allow you to have details of the birth held at the General Register Office in the UK.
This is called a Consular Birth Registration
That document is not a UK birth certificate and should not be used as one. It does not take the place of the locally issued birth certificate, which is the formal and legal certificate of birth.
Please also be aware that any registration made with the British Embassy or with the FCO here in UK does not confer British Citizenship.
There is no requirement to register the birth in order to secure a British passport.
To proceed with the consular registration for the case and circumstance you describe, you will need the following documentary evidence:-
Child’s Philippine birth certificate
Photocopy of the either the mothers passport or mothers birth certificate.
Father's full British birth certificate
Father's current British passport (or certified copy)
Paternity Declaration
If any additional information is needed the Consular dept will inform you
The registration cost is currently around £105.00
Hope this is helpful
