How much time elapses between the opening and closing lines of Part III?
At the beginning of Part III, " . . . Peyton Farquhar fell straight downward through the bridge[,] he lost consciousness and was as one already dead." After this, the narrator says that he was woken up by a terrible pressure on his neck. It is not ever made completely clear whether or not he really feels this pressure or if he only imagines it, just as he imagines swimming down the river, running through the woods, and making his way back toward his home. I believe Farquhar feels this pressure only because he expects to feel it, as the first part of his imagined escape begins with the breaking of the rope around his neck and his attempt to remove that rope once he's fallen into the water beneath the bridge. He thinks, "To die of hanging at the bottom of the river!—the idea seemed to him ludicrous."
Then, in the final line, we learn that Farquhar's neck has been snapped by the rope, and he hangs dead from the bridge. Thus, the entirety of Part III occurs in the time it takes Farquhar to begin to fall (when the sergeant steps off the plank holding Farquhar up) to when the rope stretches taut, breaking his neck. This cannot be more than one or two seconds and may, in fact, be only a fraction of a second (depending on how long the rope is). Farquhar's imagination is simply moving so quickly that it feels as though it has been a great deal longer than it actually has. We were given a clue that this could happen when Farquhar's sense of time slows down in Part I. Despite the fact that the creek was "racing madly," Farquhar feels that it is moving "sluggish[ly]," and the ticking of his watch slows down in his perception: to him, it seems as though the "intervals of silence [between the ticks of the second hand] grew progressively longer . . ."
Only a very short time elapses between those lines - probably only a few seconds. In the opening line of the section, the physical reality of Farquhar's hanging has begun, and he falls "straight downward through the bridge". In the last line, he is dead, "his body, with a broken neck, swung gently from side to side beneath the timbers of the Owl Creek bridge". Assuming that death would have occurred instantly from such a sudden, tramatic injury, all that transpires in the interim had to happen between the time he was dropped off the bridge and the time his neck snapped at the end of the rope.
It seems as though hours have passed since the rope from which he is hanging breaks. He swims through the water avoiding bullets, and runs for what seems like miles through all sorts of obstacles before he finally reaches home and is pulled into the loving arms of his wife...then BAM! He is hanging from the end of the rope again. This is a typical dream sequence type story. At the very most, minutes pass, but more likely, it's only seconds.