The fastest I know of is the heat shock locus in _Drosophila_. The transcription factor (HSF) accumulates within about 30 seconds, and RNA Polymerase can be seen to start accumulating within 3 minutes.
Katie L. Zobeck, Martin S. Buckley, Warren R. Zipfel, John T. Lis. 2010. Recruitment Timing and Dynamics of Transcription Factors at the Hsp70 Loci in Living Cells. _Molecular Cell_ 40(6): 965-975