What is a typical response time for a wired controller?

From Kotaku.com

The new proprietary protocol—to be clear, it's not WiFi nor Bluetooth, which Alam said was too slow for their needs—should reduce controller-to-console latency by 20% compared to the old 360 connection. How fast? Alam says that the time from button press on an Xbox One controller to the console receiving the signal should be 9 milliseconds. He believes that is effectively the same as with a wired controller

If the latency has been reduced by 20%, it would imply that Xbox 360 response times were around 11 ms.

This is significantly shorter than the 100-200 ms human response time you asked about. A single frame at 60 fps is 16 ms long, which is why on most games controller inputs appear to be mirrored in game instantly as the response occurs in less than one frame.