While not specifically answering when the Ents started to talk, this quote from Tolkien mentions that the elves didn't know where Ents came from, implying they actually existed before they sang to them (not such thing as Ents being awakened by Elves).
"No one knew whence they (Ents) came or first appeared. The High Elves said that the Valar did not mention them in the ’Music’. But some (Galadriel) were [of the] opinion that when Yavanna....besought Eru (through Manwe) asking him to give life to things made of living things not stone, and that the Ents were either souls sent to inhabit trees, or else that slowly took the likeness of trees owing to their inborn love of trees."
JRR Tolkien - Letter 247
It would seem strange that Ainur lost the abilitie to speak when embodied, but that's just my reasoning. However this quote seems to contradict the one you already quoted from TreeBeard in The Two Towers. Yet I'd like to add what Tolkien said about quotes from his characters:
"Treebeard is a character in my story, not me; and though he has a great memory and some earthy wisdom, he is not one of the Wise, and there is quite a lot he does not know or understand"
JRR Tolkien - Letter 153
Who talked first seems unknown, the best quote I recall about the Elves speaking is inconclusive:
"...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."
JRR Tolkien - The Silmarillion - Of the Coming of the Elves