Once or twice, while starting the live broadcast in a hurry, BUTT has shown no input. Restarting the system had been the solution. Today also I experienced the issue, but instead of restarting the machine, tried restarting BUTT and then reloading the JackPilot saved session - no go. Restarted Reaper and reloaded JackPilot saved session - that worked.So I surmise that the problem is caused when BUTT starts to broadcast before Reaper has focus, when Reaper is in the background. At that time, the Reaper output to Jack is probably not available, and later when Reaper gets focus, it does not connect to Jack for some reason. So the solution is to restart Reaper and reload the Jack session.
Edit Dec 2013: One more possible reason: if Reaper is paused - audio does not come out even from live mics through the insert send! Either REC or STOP or PLAY is needed.