You field dress the deer as soon as possible while maintaining your own safety. That really means that as soon as the deer is down, I go take care of it, except if I have shot it during the last few minutes of shooting light and I can't safely track the deer and do the work in the dark. Often I can do it in the dark if I have tracked it with enough light, but I once pushed this a little too far and had a really bad night and an extra 20 mile hike to get out. I won't do that again.
Outside of your personal safety, you should get the chest cavity cleaned out and the animal started cooling as soon as possible. If you do shoot it in the evening and are going to let it cool overnight, try to prop it up on some logs so air can get underneath especially at the hind quarters, and tie the legs out so it is on its back with the chest cavity wide-open for good air circulation. Hope that a bear or coyote doesn't discover your deer in the night. Fur is a great insulator, so you need to maximize the air getting to the un-furred places on the deer.
If you are hunting in warmer weather, you may have to work quickly to get the animal to ice, but since the answer is "as quick as possible", that means you are just going to have to do what you can as quickly as possible. You can't convince a deer that bolted and fell down a cliff that he should really have died in a better location for the haul out so you just do the best you can to be as quick as possible. :)
If the night time temps are 40 deg F, you can let it cool at night and haul it out during the day. One trick is to take an old sleeping bag and let the deer cool overnight, then wrap in the sleeping bag to keep it cool for the drive home, or if your buddies are still hunting, using the bag to keep it cool during the day can let you hang it for three or four days while they finish up. Make sure you have a thermometer to monitor the meat temp. It should be less than 45 deg F all day.
I hang mine in the shed with a remote thermometer I can watch from the house. If the temp gets up too high because of the weather, I go take the skin off and hang it in game bags in the fridge, rotating and moving it around 2-3 times a day.