What response do 'old earth creationists' have to the discovery of mature galaxies at the limits of observation?
This isn't really a Christian question.
What evidence do we have for the Universe being ancient? Well... stars are really far away. That's about it. According to materialists (that is, those who a prior reject that a divine being plays or has played a part in the universe), there is no conceivable explanation why we can see stars that are billions of light years away unless the light we're seeing is billions of years old.
There are two problems with this. First, the cosmological background radiation is more uniform than predicted by the best (materialist) models. Second, JWST is revealing that these supposedly "young" galaxies don't look any younger than much closer stars.
We have "better" evidence of "old earth". Around 1830, geologist Charles Lyell introduced the idea that "the present is the key to the past". Lyell was an atheist who, critically, denied Noah's Flood. His principle, known as Uniformitarianism, allows us to observe processes in the present day and interpolate them backwards in time to estimate ages. This has, in particular, led to the field of radiometric dating, from which we get an "age" of Earth of several billion years.
This also has problems. The same uniformitarian principle, applied to many phenomena, also yields maximum ages of certain features that are much lower, sometimes on the order of ten thousand years. Moreover, we have substantial evidence that catastrophism, not uniformitarianism, is a viable or even likely explanation for various features that are often claimed as "evidence" of great age. Ultimately, what we're left with is the fact that a lot of radioactive decay has occurred. Since we weren't there to observe it, however, the assumption that this required a great deal of time is just that; an assumption. (In fact, there is evidence that this decay occurred very quickly!)
Overall, the conclusion is that "old earth" (or, more generally, a universe that is billions of years old) is an idea which has many issues, of which JWST's findings are just a few. History clearly tells us, however, that atheists will find ways to fold, spindle and mutilate their model until, if one squints hard enough, it looks like it fits the evidence... much as epicycles were introduced to force geocentrism, or the planet Vulcan introduced to force Newtonian mechanics, to match observations.
What do we find in Scripture? Genesis 1 clearly says that "God created the heavens and the earth". It says this was accomplished in seven periods of light and darkness, of evening and morning. It says Earth was initially water, and that plants were created before the Sun, moon and stars. It gives specific genealogies that place the date of Creation at approximately 4,000 BC.
The "old earth" idea is simply not found in Scripture. It is an idea originated by God-denying, Scripture-denying men. Many Christians have chosen to place their trust in these ideas of men, believing them more trustworthy than the Word of God. Thus, there is absolutely no reason to believe that Christians subscribing to "old earth" would offer an explanation for JWST's findings that differs from atheist explanations. Rather, what we should take away from such findings is a reminder that man's ideas, such as the Big Bang, "old earth", common descent, and the like, are fallible. History is replete with scientific "facts" that have subsequently been modified or thrown out entirely.
Furthermore, as Christians, we ought to be skeptical of any claims that seem to contradict God's infallible Word. We ought similarly to be skeptical of any claims that are founded on a rejection of God and God's ability to act in Creation.
If anything, the 'old earth creationists' response ought to be repentance, and a return to the unchanging teachings of Scripture. Perhaps some will respond that way, but many others will continue to follow whatever atheistic, materialistic "science" claims.