What is the capacitor charging time when charged with a pnp current mirror circuit?
From the equation here we have 

\$ I_{ref} = I_c(1 + \frac{2}{\beta})\$,so 

\$I_c = \frac{I_ref}{1 + \frac{2}{\beta}}\$ 

where \$I_{ref} \approx \frac{12 - 0.6}{100k} = 0.114 mA\$.

From \$I = C\frac{dv}{dt} \$ we get \$ V(t) = \frac{I}{C}t\$.

Assuming the transistors in the current mirror have a high \$\beta\$ then
\$ I_c \approx I_{ref}\$.  The 555 timer upper comparator will fire when \$V_c = \frac{2}{3}V_{cc} = 8v.\$ Plugging these numbers in to the equation above along with the capacitor value and solving for t we get:

\$ 8 = \frac{1.14*10^{-5}}{470*10^{-6}}t \implies t \approx 330\$ seconds.

Even assuming that the capacitor is "ideal", high transistor beta, and that the 555 timer inputs draw no current, the actual charging time will probably be greater than this, due to the Early effect of the right hand transistor in the current mirror.