Yes, and no. The time-dilation effect will happen instantly, but the more quickly you want to observe it, the better your measurement's S/N ratio will have to

Yes, and no. The time-dilation effect will happen instantly, but the more quickly you want to observe it, the better your measurement's S/N ratio will have to be... and that, in turn, requires narrow measurement bandwidths that imply longer observation times. So then the question has to be asked, does the effect really happen instantly? Or do the same mechanisms that impose an inverse relationship between bandwidth and SNR mean that, in fact, it doesn't happen instantly at all?