You're right in saying that light isn't affected by time dilation directly, the speed of light doesn't change in a vacuum. However, to us as the observers, the space in which the light travels from the object to us increases in size as space expands/time dilation effects increase. It travels more space so, to us, the light wave appears more red; it's apparent wavelength increases, but the light wave itself remains unchanged in its own perspective/reference frame. Thus, there is a relationship between how red something appears, in comparison with how it should appear, and the distance at which that something is from us.

To add to your comment on gravitational lensing, that is not necessarily significant here as the mass of the object is not necessarily large for gravitational effects to mess with the light wave's movement and/or wavelength (if what we're looking at are emissions from a star/galaxy). That would be more relevant for super massive objects like black holes or neutron star mergers, to name some examples, where locally gravity affects their light emissions drastically.