Blinking LED Circuit – Create a Blinking LED Circuit with a Z-Diode and a Capacitor

I didn't read your whole question, which seemed to go out of its way to make a simple thing complicated. As I understand it, you have a capacitor, resistor, and LED all in series, and you want to know how things decay if the capacitor starts out initially charged up.

At first apporoximation, you can consider the LED a voltage source. That means the current will decay just as if the LED wasn't there and the cap was charged up to the LED voltage less than what it really was. This is now a simple R-C systems which follows a basic exponential decay with a time constant of RC, which it seems you already understand. The question of when the LED goes "off" then comes down to at what current you consider the brightness to be low enough to be off. This can vary a lot by the efficiency of the LED, ambient light level, and how obvious "on" is supposed to be. For example, if the cap is initially charged so that the initial current is 20 mA (a common maximum for LEDs) and you consider 1 mA the "off" level, then the on time will be the 95% decay time, which is 3.0 time constants.

As I said, this was the basic first approximation where the LED has a fixed voltage accross it. That will be largely true, but of course its voltage will drop with current somewhat. For practicle purposes, this is a small effect compared to the slop of deciding what current level "off" really is, unless that current is small, like less than a mA.

The circuit, as drawn, does not work.

Furthermore, even if it did work, it is doing so by horribly abusing the transistors base-emitter junction diode, which can potentially damage the transistor.

It's worth noting that it is possible to make the circuit work. As far as I can tell, the person drawing the diagram simply forgot a diode:

The pull-up resistor also needs to be somewhat smaller, in order to turn Q1 on hard enough that the generated current waveform is appropriately symmetric. It's 10K in the above image, but 1K works much better.