the lord of the rings - Who's older: Treebeard or Tom Bombadil?

Tom Bombadil says this to the hobbits, in reply to the question "who are you?":

(...) But you are young and I am old. Eldest, that’s what I am. Mark my words, my friends: Tom was here before the river and the trees; Tom remembers the first raindrop and the first acorn. He made paths before the Big People, and saw the little People arriving. He was here before the Kings and the graves and the Barrow-wights. When the Elves passed westward, Tom was here already, before the seas were bent. He knew the dark under the stars when it was fearless – before the Dark Lord came from Outside.

So basically, he was here before everything else. After Helm's Deep, when a party rides to Isengard, Gandalf says this to Théoden:

Well, Théoden, will you ride with me to find Treebeard? We must go round about, but it is not far. When you see Treebeard, you will learn much. For Treebeard is Fangorn, and the eldest and chief of the Ents, and when you speak with him you will hear the speech of the oldest of all living things.

Maybe this is just an inconsistency not intended by Tolkien, I don't know. A few years ago, I read somewhere that Tom Bombadil is not "a living creature" in a traditional sense, but there wasn't a strong argument supporting that. What is your take?

Answer

Yavanna had created the trees and Melkor was destroying them. So to help Yavanna, Iluvatar sent spirits with which to ensoul certain trees to serve as protectors of the others: these creatures were the Ents. Treebeard was the oldest of the Ents, and so the firstborn of all sentient or perhaps animate things on Arda, or as Gandalf loosely says, 'living creatures'. Trees and other plants did come before Ents. This is all in the Simarillion.

That takes care of Treebeard. What of Tom Bombadil?

In the entire canon of LOTR/Silmarillon (and to my knowledge all Tolkien's writings but then I haven't read them all now, have I?) there is no in-universe explanation* of who or what Bombadil is. A Maia? A Vala? Eru himself? Some other kind of being? All sorts of theories have been proposed. But in the stories? As Goldberry says, "He is".

So if Bombadil told the truth when he said he was older than the seeds of the trees, then he was older than Treebeard. But he may have lied. Or was mistaken. Or Gandalf lied. Or was mistaken. Or Bombadil may not have been of Arda but of the Heavens. The stories don't say.

*Out-of-universe, Tolkien had already invented the character of Tom Bombadil prior to LOTR and included him as "an 'allegory', or an exemplar, a particular embodying of pure (real) natural science..."