-if your bullet gets into the brain, it will usually be lights out. It has to get in there though, and as well as being surrounded by bones and sometimes headgear. it can be a challenging shot; - if your bullet causes the spinal cord, or a nerve junction nearby (such as the brachial plexus), to be severed (whether directly or by driving bone fragments through it) the animal will be down, though it might not be dead if nothing else (eg nearby arteries) is damaged; - if your bullet makes a big hole through a major artery, especially the aorta, it will usually be over pretty quick, but it might not be instant; - if you hit the heart, the pumping stops, so the animal may have a few seconds - enough to bolt into cover or at you - before going down - a hit to a major organ, other than the above, such as lungs, really needs a good deal of damage to be effective quickly - breaking major bones may drive splinters through something vital, but if not, animals can often keep on trucking
That is only my experience of course, based on animals I've killed or seen killed. Sometimes unusual things happen too.
