A typical red indicator LED will achieve full brightness in microseconds.
. Go to the web sites of a few LED makers/distributors and look at the spec sheets. Google is your friend.
The problem with this is I have no idea what my LEDs are. I get them in lots on Ebay. The OP might have the same problem.
There's no real way to test for this unless you have an oscilloscope. Though most LEDs will have similar properties, so find a datasheet for an LED of the same color and you have a good rough idea of your LED's response times.