No carole, fag and faggot are not related. The expressions "It's a fag" meaning "It's wearisome, a boring, tedious task" and "Fagged out", meaning "completely exhausted from work" show fag in its original Public School meaning for a junior boy who is required to do menial tasks. A faggot is a bundle made of twigs , or of pieces of chopped meat, bound in the shape of a roll; use your imagination ! For some strange reason it has come, in the US, to mean a homosexual.
Not sure that fagging still existed when Cameron was at Eton.