You can either obtain a 59 day visa before leaving UK or you can get one at Manila airport when you arrive.
There are a number of members who have done just that.
There is a posting here somewhere informing exactly where in the airport to secure the visa extension, but I just can't find it right now.
Hopefully those who have done it at the airport will respond.
Some folks might tell you to not bother with the visa and to just pay the overstay charges etc when you leave.
Please don't do that.