So far I've heard no confirmation that there indeed was an engine fire requiring the crew to complete an engine fire drill and emergency return.
The only fire reports are from those on the ground and likely associated with an engine surge recoverable by shutting off the fuel.
One of the problems is that the ATC archives linked above are missing the arrival/departure frequencies for the first seven minutes of the flight, and tower only has the first seven minutes. So you only hear BA 116 at the very start of the problem, and then again after the flames are extinguished. That said, here's what the tower recordings
do say:
(times are EST according to the liveatc recordings):
2323:02 BA116 Cleared for T/O
2324:09 (TWR) "Speedbird 116 Heavy you got what appears to be some flames coming out one of your engines on the left side it appears"
2324:19 "116" (loud wooping sound in the background)
...
2325:10 UA840 (from ground), notes "I think that thing is gonna burn off" (This doesn't mean, as suggested above, that the "engine is gonna burn off", but more likely "the fire is gonna burn off")
2327:20 American 314 Kennedy TWR 31R you're clear to land
2327:24 Okay clear to land I understand you still have an emergency in progress
2327:27 Yes we do
2327:31 Oh yeah, I see him he's got flames comin out
2327:34 Yeah, we know that. We told him and we're working with him
2327:37 He knows that he's got flames
2327:39 Yes
...
2328:27 TWR Two things AA 314, I understand we're cleared to land 31R and also we don't see the flames coming out of his uh left side anymore.
So whatever it was, it burned for between 3 1/2 and 5 minutes.
You may continue speculating