As a British citizen, are you allowed to enter back with a non-British passport?
Can I still use my Philippine passport to go in Philippines and back to England using it?
Paragraph 12 of the Immigration Rules says...
A person claiming to be a British citizen must prove that he has the right of abode in the United Kingdom by producing either: (i) a United Kingdom passport describing him as a British citizen or as a citizen of the United Kingdom and Colonies having the right of abode in the United Kingdom; or (ii) a certificate of entitlement duly issued by or on behalf of the Government of the United Kingdom certifying that he has the right of abode.
So a passport is required to enter the UK as a British Citizen.
If you do not, or cannot produce a British passport, then Paragraph 7 kicks in. It says...
A person who is neither a British citizen nor a Commonwealth citizen with the right of abode nor a person who is entitled to enter or remain in the United Kingdom by virtue of the provisions of the 2006 EEA Regulations requires leave to enter the United Kingdom.
That means they will inspect your Philippine passport for your UK visa (specifically, an entry clearance issued by a consular post abroad) or UK permanent residence or UK right of abode stamps. If they find a valid 'leave to enter', they will use it to admit you. The terms and conditions of that visa will apply.
If they do not find a suitable 'leave to enter' in your Philippine passport, you will be detained and you will have to think up something to explain your circumstances. Your biometrics will be taken to see if they sync up to your explanation. They are also likely to run your biometrics through Interpol and the DHS. If they are satisfied with your explanation, you will be admitted under Paragraph 18 of the rules, OR you will get a temporary admission with the condition of reporting to the police within a certain amount of time.
If they are not satisfied with your explanation, then you will be vulnerable to being served removal instructions.
It's not forbidden to use another passport but in practice the main problem you will face is that Philippines citizens need a visa to visit the UK. Without a UK visa or a passport to establish your British citizenship, you would most likely be denied boarding in the Philippines and would not even get the chance to explain your situation at the UK border.