I suspect galaxy distances are not very precise to begin with.

For example, the distance to the nearest galaxy: "The team refined the uncertainty in the distance to the LMC down to 2.2 percent." [1]

This kind of precision could be used for example to measure the radius of elementary particles (wild guess, I'm not even sure that makes sense given what we know about the quantum world)

[1] http://obs.carnegiescience.edu/content/distance-nearest-gala...

Not really. The amount of significant digits in those measurements is too small for it to be affected by such a minute redefinition of the second (and by consequence, meter). Keep in mind that we're actually changing the definition of the second (and, again, by consequence, meter); but we're still putting the new definition somewhere in the area that the old definition put it. The only reason for the redefinition is that we can't measure the things it's defined in terms of any more precisely.