Elrond says Tom was old when the Elves awoke in 'The Council of Elrond', while in The Silmarillion, Chapter 2, it implies that the Ents didn't start walking Middle-earth until the Dwarves awoke, which happened after the Elves.
When the company was taking their leave from Treebeard in Isengard on their ways back to their homes in 'Many Parting', Chapter 6, Book VI, RotK,
I found the following sentence:
And Celeborn said: 'I do not know, Eldest.'
He was speaking to Treebeard here; however, I think he may have just meant oldest of the Ents, but of course I could be wrong. And we know nothing about the age, if such a quantity can be attributed to the spirits from which the Ents and Tom Bombadil were originally formed.