It's pretty clear who will win this one. Google. Even though the word isn't a pejorative in England, it will disappear because online it's status in the US is what matters.
Quoting etymonline.com here,
The explanation that male homosexuals were called faggots because they were burned at the stake as punishment is an etymological urban legend. Burning sometimes was a punishment meted out to homosexuals in Christian Europe (on the suggestion of the Biblical fate of Sodom and Gomorrah), but in England, where parliament had made homosexuality a capital offense in 1533, hanging was the method prescribed. Use of faggot in connection with public executions had long been obscure English historical trivia by the time the word began to be used for "male homosexual" in 20th century American slang, whereas the contemptuous slang word for "woman" (in common with the other possible sources or influences listed here) was in active use early 20c., by D.H. Lawrence and James Joyce, among others.
The perjorative is the most common use, however it also describes a food and small bits of firewood. People generally seem to work out the meaning from the context relatively easily, homonyms being a thing.
To illustrate, here is a picture of a man with a giant cock - http://cdn.ecommercedns.uk/files/5/227995/1/5014991/kmk023-h...
I went to a minor public school and though fagging had been abolished many years prior, the first year of the senior school (age... 12/13ish) were still informally known as the fags. It wasn't particularly a homophobic slur, IIRC, though playground homophobia was rife back then (around 1990) so who knows really.
I do know I see "Faggot" used more for the food than as a slur these days, but then I don't hang around the *chans...
Why the verb "to fag" means what it does seems to be an open question, and doesn't seem to be related to the pile-of-sticks root.
https://www.etymonline.com/word/fag?ref=etymonline_crossrefe...
Then again, I'm a man who sleeps with other men, went to a crumbling concrete block of a state school and often can be found working in factory environments and hanging around with bikers.
I guess it varies greatly according to one's milieu.