The collector current is constant only if there is nothing else to limit the current. If the available collector current is more than the current through P1 (with T1 acting as a saturated switch), then the voltage at the collector is constant and the current is determined by the voltage drop across P1. This voltage decreases as C4 charges, thus the charge current to the capacitor is exponential.
If you look at the collector voltage in your simulation, you will see what I am saying.