The multiplexer transistors switching time is less than 1us. It depends not only on the multiplexer switching (which is really small), but also internal load capacitance charging, analog buffer setting time, etc. These values limit the upper bound of switching time.
With the software intervention (ISR), the amount of time is based on the amount of time between the IRQ pin being asserted to the firmware instruction that changes the AMUX register. This time is dependent on CPU clock cycles and therefore your CPU clock. It takes the M8C interrupt controller about 20 CPU cycles from interrupt detection to code execution in the ISR. The amount of time in your ISR is dependent on the code that you have written for the ISR. If the multiplexer switch is the first instruction in the ISR, you need to add 8 CPU cycles to the above 20.