Calculate charge time of a capacitor through a current limiting power supply?
As it is not a piecewise continuous system, you'll need to calculate, in two steps:

First: you'll need to calculate the time of charging the capacitor until it reaches

$$ (Vb-Vc)/R = Imax $$

with constant current of Imax.  if the current is constant that capacitance does not change this is a simple straight ramp curve upto the point where the current is no longer limited by the constant current.

Second: you can use the equations for power supply without limited current power supply.

Which are the exponential curves from that point onward to the asymptote.