RC time constant changes with introducing LED
An LED is not a resistor! Simple as that. 

It is a totally non-linear device. Look for I/V curves of diodes to understand what I mean:


(by H1Voltage from Wikimedia Commons)

A resistor would be a perfect straight diagonal line through the origin point with the slope being the resistance.

That means that for low voltages, the "apparent" ohmic resistance is large, and for high voltages very low. For negative voltages (which can be often involved when one is concerned with things like step responses), we typically even model diodes as perfect isolators (R going to infinity).

Especially, that means that you can't just go and say "hey, that's the time constant of my R-C-LED circuit", since unlike RC circuits, you can't just scale the input voltage and get a simply scaled output. A diode is a nonlinear device!

So, most probably that means you're trying to calculate/simulate/measure a number that's practically meaningless.