its not the phils gov that will be giving him a visa to live in the uk, its the British embassy, so phil law has little to do with it
sure she might have custody and sole responsibility, but if his father/family protested about the son leaving to the embassy it could cause some problems. last thing the embassy want is to be stuck in the middle of a fight btw 2 parents or families..
is the father mentioned on his birth cert ?
you mention he sees his son how often ?