Instantly more or less. Time instantly moves differently at altitude because you are in a weaker gravitational field. The time dilation effect would be noticeable after 1 (or at most a few) ticks of the clocks.
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?
Let's imagine that there is a huge amount of time dilation (we live on the surface of a neuron star or something). By climbing a bit, we experience 1.1 seconds instead of 1.0 seconds experienced by someone who left down.
We have a clock that can measure milliseconds as the smallest tick. But climbing up, back down, and comparing the amount of ticks won't let us conclude anything after a single millisecond. If anything, we must spend at least 11 milliseconds up to have a noticeable 11 to 10 millisecond difference.
Now, if the dilation was 1.01 seconds vs 1.00, we would need to spend at least 101 milliseconds up, to get a minimal comparison between 101 and 100 milliseconds.
That idea is the premise of https://en.wikipedia.org/wiki/Incandescence_(novel)
Nevertheless, in order to measure a frequency difference between two optical clocks you do not need to count their signals. The optical signals can be mixed in a non-linear optical medium, which will provide a signal whose frequency is equal to the difference between the input frequencies.
That signal might have a frequency no greater than 1 GHz, so it might be easy to count with a digital counter.
Of course, the smaller the frequency difference is, the longer must be the time used for counting, to get enough significant digits.
The laser used in this clock has a frequency around 200 THz (like for optical fiber lasers), i.e. about 2E14 Hz. This choice of frequency allows the use of standard optical fibers to compare the frequencies of different optical clocks, even when they are located at great distances.
Mixing the light beams of 2 such lasers, in the case of a 1E-17 frequency difference would give a difference signal with a period of many minutes, which might need to be counted for several days to give an acceptable precision. The time can be reduced by a small factor selecting some harmonic, but it would still be of some days.