If I am not mistaken,
Gandalf says that
Treebeard is "the oldest of
living things". This is not to mean that
Maiar are dead

, I suppose, but all
Ainur are older (since created before
Arda) than Treebeard -- including of course Gandalf himself. So why the qualification? I would think it more natural to say oldest of living
beings -- a subtle distinction, but important. A thing, as opposed as a being, has materiality, occupies a position in space, etc.
My interpretation is that Gandalf is saying that,
among the Children of Ilúvatar, Treebeard is the oldest. The Children include
Dwarves (whom Ilúvatar adopted as His own after
Aulë's creation) and
Ents (created at the bidding of Yavánna). There is no children older than Treebeard -- no
Elves, or other Ents (the only immortal children). This does not mean Treebeard was around since the beginning of time, or is a
Maia. He is simply the oldest one around. (Treebeard seems to have some memory of being an Enting, doesn't he? I'm not sure about that, gotta check it. I assume this would mean he had a father and a mother -- or did the children come to be in a immature body? Someone with HoME info could help us on that).
Bombadil, on the other hand, claimed to be around before the first rain fell, and before the first plant grew. This seems to indicate that he was here since the beginning (remember, the first rain fell right at the beginning of the world, as
Melkor used fire and cold to hamper the work of the other
Valar -- the origin of clouds and snow).
My conclusion: Bombadil is the oldest (in fact, his words can be interpreted to mean that he was the first spirit to enter Arda). Treebeard is the oldest surviving child of Ilúvatar.