Ents are likely older than Elves
On at least two occasions, Tolkien notes that the Ents are the oldest sentient race in Middle-earth (bold is my emphasis, italic is Tolkien's):
That is a long and yet bald resume. Many characters important to the tale are not even mentioned. Even some whole inventions like the remarkable Ents, oldest of living rational creatures, Shepherds of the Trees, are omitted.
The Letters of J.R.R. Tolkien 131: To Milton Waldman. 1951
Ents.The most ancient people surviving in the Third Age were the Onodrim or Enyd. Ent was the form of their name in the language of Rohan.
Return of the King Appendix F I: "The Languages and Peoples of the Third Age" Of Other Races
Appendix F goes on to suggest that the Elves awoke in Ents the desire to begin speaking, though the Entish language was of their own devising:
They were known to the Eldar in ancient days, and to the Eldar indeed the Ents ascribed not their own language but the desire for speech.
Return of the King Appendix F I: "The Languages and Peoples of the Third Age" Of Other Races
The exact relationship between the Ents and the Huorns is indeterminate from Tolkien's writings1, but it's clear that they are related, and were almost certainly created at the same time.
The Elves, for their part, are implied to have begun speaking (and singing) some time after their first awakening, not literally from their first moments:
Long they dwelt in their first home by the water under stars, and they walked the Earth in wonder; and they began to make speech and to give names to all things that they perceived. Themselves they named the Quendi, signifying those that speak with voices; for as yet they had met no other living things that spoke or sang.
The Silmarillion III Quenta Silmarillion Chapter 3: "Of the Coming of the Elves and the Captivity of Melkor"
1 Merry, of course, believes that Huorns are Ents that have "gone treeish"; but he, like Treebeard, is merely a character and not omniscient