The Dwarves were not part of the Music, so Eru did not give them life until after his discussion with Aulë.
But Ilúvatar spoke again and said: ‘Even as I gave being to the thoughts of the Ainur at the beginning of the World, so now I have taken up thy desire and given to it a place therein; but in no other way will I amend thy handiwork, and as thou hast made it, so shall it be. But I will not suffer this: that these should come before the Firstborn of my design, nor that thy impatience should be rewarded. (The Silmarillion)
On the contrary, animals and plants were present in the music, so they were given life by Eru:
Then Manwë awoke, and he went down to Yavanna upon Ezellohar, and he sat beside her beneath the Two Trees. And Manwë said: 'O Kementári, Eru hath spoken, saying: “Do then any of the Valar suppose that I did not hear all the Song, even the least sound of the least voice? 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. For a time: while the Firstborn are in their power, and while the Secondborn are young.” But dost thou not now remember, Kementári, that thy thought sang not always alone? Did not thy thought and mine meet also, so that we took wing together like great birds that soar above the clouds? That also shall come to be by the heed of Ilúvatar, and before the Children awake there shall go forth with wings like the wind the Eagles of the Lords of the West.’ (The Silmarillion)
The clear implication is that Yavanna's work was made to Be by Eru at the time of the Music, including the kelvar and the olvar (animals and plants), and that Eru is here reminding her that also included in her Song was habitation by spirits in animals and trees that would be able to defend them from the depredations of the Children. So as the Dwarves were made by Aulë, the form of the animals and plants were made by Yavanna, but they were given life by Eru - it was just ordained during the Music rather than well afterwards, as with the Dwarves.