The Psalms can be divided into 5 books marked by benedictions. For instance, Book 1 and Book 2 are separated by a verse that is usually included as a conclusion to Psalm 41:
Blessed is the LORD, God of Israel, from eternity to eternity. Amen and Amen.
So Psalm 14 is in the first and Psalm 53 is in the second book of Psalms. Psalm 40:14-18 (or 13-17 in most Christian translations) and Psalm 70 are also duplicates across those same books. This gives strong evidence that the two collections of psalms circulated independently before they were compiled into the current collection of 150 poems. So in many ways the question is which collection came first. Psalms 40 and 70 can also be used as evidence. (For a side-by-side comparison, see this gist.)
One notable difference between 14 and 53 is the later's reluctance to use the Tetragrammaton. In modern Judaism, there are prohibitions against speaking the name of God or writing unnecessary. It seems possible a version of the prohibition was developed before the Psalms were collected, which would indicate Psalm 14 is an older version. Indeed, the second book frequently avoids the name of God by substituting another word (often 'elohiym) where you might expect LORD to appear. So, for instance, Psalm 43:4 uses "O God, my God" (אלהים אלהי) which seems overly repetitive. Psalm 7:3, to pick a random example from the first book, reads "O LORD my God" (יהוה אלהי), which seems more natural.
Still, it doesn't appear the editor of the second book of Psalms mechanically elided the Tetragrammaton. Psalm 70 ends with "O LORD, do not delay." Meanwhile, Psalm 40 ends with "O my God, do not delay." So at least in that instance, the first and second books swapped preferences for how to address God. We also can only speculate why one name was preferred over the other. Perhaps the reason for using LORD was to distinguish the deity being addressed from the Canaanite El Elyon. In other words, it seems possible Psalm 53, which does not use LORD, is the older text.
Another clue is the addition of machalath in the superscription. We don't exactly know what it means, but it seems likely it's an aid to the musicians. While that could mean an editor removed this detail from Psalm 14, it's hard to know why. Scribes tend to be conservative in what they copy so it seems unusual for a word, especially a rare word appearing just twice in the Psalms, to be dropped altogether. In addition, Psalm 53 also adds maskiyl a far more common word to describe Psalms. A simpler explanation is that Psalm 14 was written down at a time when it was regularly performed so the additional musical instructions were not needed. It's interesting to note that Psalm 70 also adds more detail to Psalm 40's superscription, though it is not a musical instruction.
The divergent text near the end of the Psalms changes the entire tone. Psalm 14 puts God in the position of defending the godly, while Psalm 53 has him annihilating the people who are attacking Israel. (The literal phrase is “scatters the bones”, which is an image of utter defeat.) It's not wise to pick specific historical events to which these lines refer. At best we can speculate that "defending the godly" might have less impact when the audience is already defeated by their enemies.
On the whole, it looks as if Psalm 53 reworks Psalm 14 rather than the other way around. The substitution of "God" for "LORD" and the additions to the superscription suggest 53 is further from the oral tradition. To my reading, 53 is also more focused on violent revenge, which seems more fitting to a period when the Hebrew people were suffering military and political defeats. The simplest explanation is that Psalm 53 was a copy of Psalm 14.