How responsive are current transducers?
This is not really easy to say, because the device you are linking to is an integrated product, not the raw current transducer. In other words, besides the current-sensing component, there is likely a microcontroller or RMS-to-DC converter in the device. This device integrates the current signal over some amount of time and then outputs a 4-20mA or 0-5VDC signal proportional to the RMS current that has flown through the transducer.

To make this answer a bit more general: whether they are using a hall-effect sensor or a current transformer internally, the bare sensor has an instantaneous response (sub-microsecond even with a compensated current transformer topology). However, such a signal is not compatible with 4-20mA or 0-5V because the current signal is AC. Therefore, the manufacturer has built something that integrates the current signal over at least one entire current cycle (20ms for 50Hz), but judging from the marketing material likely more cycles. This is the main source of the lag from input to output.