In my informal experience, I have found that the caffeine portion of coffee tends to kick in very quickly. The transition from coffee to bloodstream to brain doesn’t seem to take more than a few minutes.
This morning I had to give my wife a ride to work after only sleeping two hours. When I first got up I could barely walk down the hall. After one cup of coffee and a couple of minutes I had almost 100% of my mental facilities up and running to drive there and back without a hitch.
A lot of this might be psychological, of course. But still, this has been my consistent experience.