I don't think it a case of it taking a long time to measure the signal, I think it's more that if they did it any quicker you would notice it happening (in the audio output of the station you were listening to).

As I understand it the tuner can only be tuned to a single freqeuncy at a time, so they would have to quickly switch to another frequency and then back to the one you are listening (presumably during the switch the audio output it muted or held at what it was before the switch???). They can only do this quick switch so many times a second before it becomes noticable in the audio output.