I am assuming you haven't already legally formed a civil partnership in a country (not the Philippines) that recogises same-sex marriages/partnerships. You can't register as civil partners at the British embassy in Manila as such. What you can do is apply for a visa as a proposed civil partner. Assuming this visa is granted you can enter the UK and within 6 months you must have a civil partnership ceremony and then applied for another visa.
To apply for a visa as a proposed civil partner you must show evidence of your relationship and that your sponsor can support you in terms of money and accomodation.
Providing a CENOMAR and a Certificate of No Impediment is not essential when applying for a visa as a proposed civil partner.