A faggot originally meant a bundle of sticks or twigs, and they were used to light fires, but I don't think this has any relation to "fag" as in cigarette. Etymonline says of the latter:

British slang for "cigarette" (originally, especially, the butt of a smoked cigarette), 1888, probably from fag "loose piece, last remnant of cloth" (late 14c., as in fag-end "extreme end, loose piece," 1610s)

That meaning of faggot, interestingly, comes from the same root as the Roman symbol "fasces" which is a bundle of sticks from which we get the modern word fascism.

Another fun fact: there's a traditional British dish called faggots which are a kind of meatball made from offal, somewhat similar to haggis but uncased.