Implementation of Adjustable Current Limiting Circuit
The problem you are hitting is common to all voltage sources that have current limiting.

Consider a lab power supply, where you've dialed in 10V, 100mA.
You haven't connected anything to the output posts, so it is in CV mode, holding the output at 10V.
If you now connect a 1Ω resistor, the lab supply should quickly switch to constant-current mode; however this cannot happen instantaneously. There will always be some capacitance present at the output, the PSU needs it for stability. Lab PSUs are typically designed to have low capacitance outputs for this exact reason, but as said some capacitance is unavoidable. This capacitance quickly discharges through the 1Ω you connected, so the load (= the resistor) receives a current that exceeds the limit many, many times, for this brief moment.

Your circuit does not exhibit this specific issue, however it is of the same general kind: the voltage at the output of a power supply cannot change instantaneously, so some brief overcurrent should be acceptable, and I don't think 1ms is that bad at all.

Your comments that you also have a voltage limiter (not shown in the circuit) hints at that you've designed your own linear regulator, too. You can ask this as a separate question, I suspect that part can be improved. The current limiting part is not bad as far as reaction time is concerned. 

I would be more worried about the oscillations you have until the current stabilizes though - the feedback loop needs more dampening.