So, the slur goes back to the now mostly archaic 'fagot' for a bundle of sticks or twigs, which is itself derived from an even more ancient word. The Latin word fascis, which the Romans called the bundle of sticks magistrates carried to symbolize their state power, descended from that same ancient root. English apparently did not borrow fagot from Latin. It probably came down to us from proto-Germanic, or possibly as a borrowing later on from one of our Germanic cousin languages.

Interestingly, the Latin word came to be associated with the exercise of state power, and is where "fascist" comes from. But the English word became associated with brooms of all things. Sweeping up being women's work, fagots became associated with women.

And guess how you insult a gay man? Nobody knows exactly when fagot became associated with men who prefer sex with other men, but it's not attested until sometime in the mid to late 18th century in New England. Sometime around that time, the verb "to fag" was used to say "do women's work."

Speculation is that people would use that verb about effeminate men, and that the fagot (the bundle of sticks used as a broom) took on the meaning too.

To fag was later used as a verb in the elite British private school system to describe younger boys being forced to do menial labor for older boys. This may have reinforced the use of "to fag" as a homosexual slur, because it wasn't only menial labor that the younger boys were supposedly forced to perform. Ahem.

But the British fagging system can't have been the origin of the slur, because it evidently started in New England and didn't become used (and never really became popular) across the Atlantic until many decades later.

So I guess we Americans get to take the credit, at least mostly.

The Brits did start using fag as slang for cigarette, evidently because a burning cigarette looks like a burning twig, and I guess it seemed reasonable that if a fagot is a bundle of twigs, then one twig is a fag.

So there you have it. Fag as a cigarette evolved very differently from fag as a slur, and the slur arose in North America well after the witch burnings of the middle ages folk etymology suggests was the origin.

Fagot is also related to fascist, in a strange twist, but in an stranger twist, the delectable fajita is also cognate.