How much time elapses between the opening and closing lines of Part III?
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.