I'm not an expert so take what I say with a grain of salt, but for example 1 meter is defined to be:
"Length of the path travelled by light in a vacuum in 1⁄299,792,458 of a second (17th CGPM)" [1]
So if you manage to measure the second with better accuracy you also manage to measure distances with less uncertainty.
For our everyday lives it will probably make no difference if the uncertainty of the meter changes from 10^-10 to 10^-11, but physicists could probably use the improvement to do something interesting.
Also not an expert but I imagine if that were the case, it would have applications in measuring distance between galaxies and other areas of enormous distance.
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)
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.
This is really just off the cuff, and I haven't done the math and all that, but various forms of real (not simulated) imaging are entering increasingly small scales and timeframes.
Clock accuracy pertains directly to such imagining. I, um, "imagine" there is relevance to this work.
If the new second is found to be longer than the old second, then for example long distance travels won't take as many seconds than before. Oh wait... sorry.
