The appropriate power diodes and low ESR electrolytic cap will have an ESR*C time constant of <10~100us while cheap e-caps rated for much lower ripple current and higher ESR will be 200uS to >1000us.

So it is not instantaneous but fast. Lower ESR caps are always preferred due to I^2 R heat loss.

Meanwhile the load R * C time constant must be chosen for a reasonable ripple level such as 10% where it works out that RC must be>= 16/f for line frequency at bridge input.  This results in very large peak currents at full load equal to % Vripple in duty cycle (d) and 1/d in peak charge currents.

Also,with at no load the caps will rise to 1.4*Vac rms at full load and also rise another 8~10% from Vac drop at rated load or in other words the Vdc can swing as much as 50% higher with no compared to full load.

This is why we call it an unregulated power supply.