I have a Philippine passport but currently residing in the United States as a greencard holder. I am travelling to Norway with a Shengen visa but I have a 23 hour layover in London Heathrow. And I plan to meet a friend and do a little bit of sightseeing. I don’t have enough time to apply for a Transit visa due to time restraint, is it possible to get one when I arrived at the airport?
You cannot get a visa at the airport, but you seem to qualify for the landside Transit Without Visa scheme. This is open for you since you hold a valid permanent residence permit issued by the United States, and you leave by air on the next day after you arrive.
To use this scheme, simply join the immigration line for non-EU/EEA/Swiss passports, and explain to the Immigration Officer that you're in transit. You'll want to have your green card ready too.
You'll need to convince the IO that you are "genuinely in transit to another country" and "genuinely intend and [are] able to" take your onward flight the next day. So be prepared that you might have to explain e.g. what you're going to Norway for. But overall that's just the same kind of burden-of-persuasion that visa-free visitors need to meet.
See the formal conditions in paragraphs V7.6 through V7.9 of the Immigration Rules.