Get it sorted out legally in the Phils first, then you can truthfully complete any British forms with the correct supporting documentation.
Simply point out to the Phils authorities that for some (unknown

) reason the birth details were wrongly filed, and you would now like to correct the error and apply for a new birth certificate.
Attempt to do this first without (expensive) legal assistance, and should that present problems, then seek legal advice.
Ultimately a DNA test will prove your relationship, if necessary.