It just varies tremendously. When I broke my back, I was immediately incapacitated. But I've also been incapacitated by the very minor and temporary effect of being winded, while, when I broke my neck, I was walking around in shock and probably could have walked home.
My point is that it just can vary a lot, so I personally don't see the need to delay. Even minor injuries can really mess you up, while major ones might not necessarily really affect you immediately, depending on a whole host of factors. I would not assume that all injuries are delayed due to adrenaline, though; some will really mess you up physically and/or mentally immediately.