Treebeard as an Ent was created as part of the Music of the Ainur, and the awakening of the first Ents is precisely datable to the same time as the Elves awoke, as is told in the Silmarillion chapter Of Aule and Yavanna:
'Yet it was in the Song', said Yavanna. 'For while thou wert in the heavens and with Ulmo built the clouds and poured out the rains, I lifted up the branches of great trees to receive them, and some sang to Ilúvatar amid the wind and the rain.'
Behold! When the Children awake, then the thought of Yavanna will awake also, and it will summon spirits from afar, and they will go among the kelvar and the olvar, and some will dwell therein, and be held in reverence, and their just anger shall be feared.
This means that Treebeard's origin is at the very earliest Year of the Trees 1050, just immediately after Varda's second star-making. If we take Tom's claims at face-value:
- Tom was here before the river and the trees
- Tom remembers the first raindrop and the first acorn.
- This dates him to at the latest the time of the Valar's dwelling on Almaren and the time of the Lamps, because during that time (Silmarillion, Of the Beginning of Days):
Since this was before the Years of the Trees, Tom Bombadil is therefore oldest (although whether he - whatever he is - could be called "living" is another matter).