We don't know exactly, but it took the elephant about half an hour (or so Orwell was told) to die after Orwell shot it. We do not really have any very good idea as to how long it was between the time he got the gun and the time he shot the elephant.
It seems likely that it was only a short time. Orwell knew that he had to act quickly so that he would not look weak in front of the Burmese crowd. So that makes it likely that he just got the gun, walked "a few hundred yards," sized up the scene, lay down and shot the elephant.
So it seems likely that it was something like 30 to 45 minutes all told.