I don't quite understand this. It kinda has to, at least during the transition from a lower time delay to a higher time delay, and vice versa. Though I would definitely accept that it doesn't cause any change in wavelength along a path that has no change in the time delay.
But in response to the OP, no, the Shapiro time delay can't realistically have anything at all to do with the redshift of galaxies, because if it were then you'd end up with dramatically different redshifts just because one galaxy happened to be behind another, and since as many galaxies are going to be transitioning into such an alignment as transitioning out of that alignment, there would be exactly as much redshift as blueshift. None of this remotely matches what we observe.
However, the Shapiro time delay is useful in cosmology, and attempts have been made to use this time delay to, for example, measure the Hubble expansion rate. The basic setup is that they observe a lensed object which has two or more images, an object which has some sort of variation (quasars, for example, vary a lot, and a lot of lensed objects are quasars). In general, we expect that different paths that the photons take will have different travel times. So we might see one image brighten just a few moments before another image, and the difference in travel times gives us information about the curvature of space-time around that lens.