Time constant of a circuit involving resistors, capacitors ... and diodes

[drdm]

Hello all,
I came across an issue that I seem to be stuck on.
I am trying to calculate the time constant of the attached circuit, but I don't know how to approach the problem.
I want to find the voltage of OUT at a given time after the PNP is switched.
Obviously I can simulate it on LTSpice, but i also need an equation.
If I had only resistors the RC time constant is obvious - C1*(R1||R2||R3||R4). But the diodes throw me off. How do they
affect the equivalent resistance of the circuit while charging the cap and how can I derive it in an equation?
Any hints will be most welcome!

[PGPG]

I've just registered to EEVblog. Don't know how to work here - for example when I started reply I see only your text, but not your schematic.

Everything depends on what diode model you want to use.
If these are ideal diodes with 0V drop on them then they conduct at once and you have circuit like there were no diodes.
Next model you can use is the diode that start conduct at specified voltage. Then a question - these are germanium or silicon diodes?
If silicon then assumed voltage will depend on expected current. But we don't know R. If current will be low (below 1mA) I would assume 0.6V as a moment when diode starts conduct. For higher currents 0.65V and for close to 1A may be 0.7V. I will use Ud as a symbol for voltage when diode starts conduct.
At the beginning with time constant RC/2 capacitor is loaded from 0V to 5V. But when voltage reaches Ud it starts to be loaded from Vd with RC/3 time constant toward end voltage (10+Ud)/3. Then when voltage reaches 2Ud it starts to be loaded with time constant RC/4 toward end voltage (10+3Ud)/4.
If you want to use nonlinear diode model then it is easier to use Spice 

[TimFox]

Since the currents are not linear functions of the voltages, the term "time constant" is not applicable.
Instead, you need to write (or graph) the output vs. time for different applied voltages as a family of curves.
A SPICE simulation is probably the practicable approach.