How fast can an F-22 scramble?
For military aircraft on "alert", there is a position called cocked on (or as some news sites call it, "hot-cocked").  This procedure is run before the aircraft is placed on alert, and does a pre-flight check of all the instruments and then places them in a state that allows power application in a rapid manner.  This way, only minimal steps are required for the aircraft to be ready.  In addition to cocking on the aircraft, they can be parked in special areas (some are called Christmas Trees because of their appearance). This allows the aircraft to reach the runway quickly. Also, in the old SIOP days, there was an Alert Facility near where the aircraft were parked, where the crew would stay in a state of readiness.



Currently the full SIOP mission isn't active (it was ended in 1992 by President Bush, but there are still forces dedicated to it in a new capacity). However, the facilities still exist on many bases.  They can use these facilities still for whatever alerts are required.

With an aircraft in cocked on position, with a pilot on alert (cockpit or facility), an aircraft can take off in a very, very short time (from seconds to minutes).  In general, the mission requirements will dictate the amount of time that the pilot is given to take off (as you note in your question).  So sadly, I can't give a specific answer to your question, because it depends on parameters you haven't specified.