According to USDA Guidelines, meat should be refrigerated within two hours. Obviously a gunshot wound is far different from exposed meat, but the principle is the same - get the meat cold as soon as possible. The skin is going to act as packaging for the meat, but you don't want to leave it out all day.
Once the blood has stopped pumping, there is nothing keeping the bacteria at bay. The "danger zone" for meat is 40 degrees - 140 degrees Farenheit. If you're hunting at night in the winter or fall, this may buy you all night long - but within a few hours of sunrise, you'll need to protect it.
When I've dressed my own turkeys and chickens, I make sure that the entire process - from slit to pluck to evisceration - is done within three hours.