Which characteristic of a BJT determines the timing in the following given application?
Transistors are not perfect switches.  They have parasitic capacitances.  When a base current is applied, some of them charge up and and when base current is removed, some will discharge (\$Cπ\$).  

A similar behavior happens when a voltage is applied to the collector (\$Ccs\$).
This takes time, and thus the RC curve you see in your graph.

Here is an small signal model of a bjt transistor 
from this question


There are three capacitors in it.  Try building your circuit with this model and see which capacitances most affect your graph.

UPDATE:
specific to your circuit, with the base at a constant 2mA and no collector voltage \$Vc\$, then \$Ccs\$ will obtain some amount of charge.  Then when the switch closes a higher voltage is applied to the collector and \$Ccs\$ will charge via \$rc\$.  This charging/discharging \$RC\$ is most likely what you are seeing.