I would think that would also depend upon where you are hunting. I've read that in places like Pennsylvania, the woods are thick with hunters during the season. Apparently, waiting any length of time practically guarantees that when you find the critter, it will have a couple more holes in it and somebody's tag already applied. I have no experience in hunting areas that are like that, and don't really want any.
I hunted two countries in Africa, shot a total of a dozen or so critters, and in every case the PH, the trackers and I were literally jogging after the animal immediately after the shot. There were certainly no other human hunters in the area, but plenty of hungry critters looking for an easy meal, so the SOP there was to not waste any time before pursuing.
Deer here at home? I'd have to agree that it varies with each individual instance, depending upon how the shot felt to you, how the animal reacted to the hit, the terrain, etc. I hit a small buck a number of years ago who ran a short distance into a patch of small poplar very late in the day. Everything about the hit and his reaction to it told me to wait, but...I was hoping to find it before it got dark. I immediately walked all around the thick patch and ascertained that the deer hadn't come out the other side. Going against my better judgment I pushed in slowly on the far side, opposite to the direction from which I had shot. After only a few minutes of very slow going, I heard the critter crashing out towards the open field. I rushed forward as quickly as possible and got to the edge of the thick stuff just in time to see the animal at the far side of the cleared field, and was fortunately able to knock him down before he disappeared into the much larger wooded area on that side.
If I had waited as I knew I should, I'd positive that the deer would have just quietly died in there, based upon the first hit. There was snow on the ground, and it would have been fairly easy to follow him a half-hour later and find him, even in the dark, but impatience got the better of me. I ended up with a nice fat little table buck who got a surge of adrenalin upon being shot, and then another dose when I kicked him up from his deathbed a short while later. Not the tastiest venison I've ever had.
Never again. Trust your own judgment and instincts and don't rush things due to impatience.