Ignoring the GSW, a person drowning will assume a vertical position, with just the top of the head floating. As muscle control disappears, and the person becomes closer to death, the body will gradually obtain the position shown in the picture. But by gradual here, I'm talking two to five minutes. Very little of the body is going to be above water level. Just the back of the head, maybe the tops of the ears.
Very few lean people float well. It involves being able to remove tension entirely from your muscles, which isn't easily done. Fat people usually float better. And you'll find you float differently in the ocean than a lake. Swimming pools usually increase buoyancy, although not as much as the ocean.
The more I think about, I'm not sure the GSW would effect the drowning process, other than speeding it up.
If you fall into the water backwards and relatively flat and very dead, your body will gradually rotate into a face down position. And this is with a significant rotational force.
So much here depends upon a combination of how close to death the guy is when he goes into the water and how calm he is in water. When we're disoriented, we tend to panic and panic increases the drowning factor.
My impression is that they're going to discover the body within five minutes after it enters the water. So he's shot and drops into the deep end of the pool, with heavy shoes. He's still alive in the technical sense, but virtually non-responsive from the GSW. Body falls on its back as it hits the water, but as the body goes down from the fall, the feet sink a little quicker and the body obtains a vertical orientation. Probably the body will continue downwards until the feet reach the bottom.
At that point, there may be some instinctive push to the surface, there will also be some compression of the body which will then cause a bit of a spring, and there's also a natural tendency to come to the surface. As the victim dies, both from the GSW and the drowning, the body will start bobbing, but still maintaining a vertical orientation. Not much of the body will be clearing the surface.
At about five minutes after entering the water, the body should be dead, although possibility of some remaining heart function. Or in other words, CPR could work. Of course, with a GSW to the chest, chances are next to impossible. Position would be relatively vertical. Eyes would probably be at water level, and as the body bobs from the motion of the water, go above and below the water line. GSW would still be leaking air and blood.
Gradually over the next three or four minutes, the body will rotate into a face down position, with a vertical orientation probably around 60 degrees, or more vertical than normal.
Best of luck,
Jim Clark-Dawe