I think there's two possibilities:
Eru himself took Gandalf at the moment of his death, enlightened him further, and sent him back.
Sauron and Saruman did go to the Void, and we didn't see Gandalf's spirit in the text, and it wasn't depicted in the movie.
In the case of (1), we have the following statement from Letters:
That I should say is what the Authority wished, as a set-off to Saruman. The ‘wizards', as such, had failed; or if you like: the crisis had become too grave and needed an enhancement of power. So Gandalf sacrificed himself, was accepted, and enhanced, and returned. ‘Yes, that was the name. I was Gandalf.’ Of course he remains similar in personality and idiosyncrasy, but both his wisdom and power are much greater. When he speaks he commands attention; the old Gandalf could not have dealt so with Théoden, nor with Saruman. He is still under the obligation of concealing his power and of teaching rather than forcing or dominating wills, but where the physical powers of the Enemy are too great for the good will of the opposers to be effective he can act in emergency as an ‘angel’ - no more violently than the release of St Peter from prison.
Gandalf was not merely sent back, he was fundamentally changed; and before he could be enhanced or sent back, he was 'accepted'.
Tolkien then tells us explicitly that Gandalf left the World, and that it was Eru, not the Valar, who sent him back:
He was sent by a mere prudent plan of the angelic Valar or governors; but Authority had taken up this plan and enlarged it, at the moment of its failure. ‘Naked I was sent back - for a brief time, until my task is done’. Sent back by whom, and whence? Not by the ‘gods' whose business is only with this embodied world and its time; for he passed ‘out of thought and time’. Naked is alas! unclear. It was meant just literally, ‘unclothed like a child’ (not discarnate), and so ready to receive the white robes of the highest.
Though it is not stated, we could infer that when Gandalf 'strayed out of thought and time', that was part of Eru's miraculous intervention that changed him and brought him back to life, if indeed the spirits of the Ainur are not capable of leaving Time on their own, incarnate or not.
But there is some doubt whether or not Sauron and Saruman did leave the World or not. Sauron's death is described as you say: a great spirit appears in the air, and a cold clear wind from the West dissolves it away.
But on the other hand, in the Valaquenta, we are told this:
Among those of his servants that have names the greatest was that spirit whom the Eldar called Sauron, or Gorthaur the Cruel. In his beginning he was of the Maiar of Aulë, and he remained mighty in the lore of that people. In all the deeds of Melkor the Morgoth upon Arda, in his vast works and in the deceits of his cunning, Sauron had a part, and was only less evil than his master in that for long he served another and not himself. But in after years he rose like a shadow of Morgoth and a ghost of his malice, and walked behind him on the same ruinous path down into the Void.
Then Sauron did end up in the Void - assuming you interpret the Void to be the Void that exists outside the World, and not merely Middle-Earth. And if Sauron went, after the description of his death, it's not hard to imagine Saruman going too. Perhaps it is the fate of Ainur who have become otherwise totally 'bound' to the world and 'incarnate', whether wickedly, like Morgoth and Sauron, or with good purposes, like the Wizards.
Another interesting passage, from LotR, is this:
Get out, you old Wight! Vanish in the sunlight!
Shrivel like the cold mist, like the winds go wailing,
Out into the barren lands far beyond the mountains!
Come never here again! Leave your barrow empty!
Lost and forgotten be, darker than the darkness,
Where gates stand for ever shut, till the world is mended.
At these words there was a cry and part of the inner end of the chamber fell in with a crash. Then there was a long trailing shriek, fading away into an unguessable distance; and after that silence.*
The Wights were probably the spirits of Men, but it seems that Bombadil literally expelled the Wight from Arda to outside into the Void, where "the gates stand forever shut, 'til the world is ended", and the process somewhat resembled the Wight being invisibly "blown away."
The trouble with that interpretation is, in Morgoth's Ring, when Tolkien re-cosmologizes the universe, he introduces us to some confusion: Elves and Men may write Void, thinking of the place outside Time, but that the actual referent may in fact be normal, empty space. The implication is also that the Ainur cannot leave the world no matter what without the intervention of Eru.
We read that he was then thrust out into the Void.(10) That should mean that he was put outside Time and Space, outside Ea altogether; but if that were so this would imply a direct intervention of Eru (with or without supplication of the Valar). It may however refer inaccurately * to the extrusion or flight of his spirit from Arda.
Now, the applicability of that statement is somewhat question - Tolkien never rewrote the whole of the Silmarillion and the Lord of the Rings to make sense within the re-cosmologized universe. But it is, perhaps, still something to keep in mind.
So ultimately, the answer is unclear, but I think the easiest and most parsimonious explanation is simply to take a wide view of (1) and say Eru did it, but I wouldn't rule out the idea that at that time in Tolkien's conception, embodied Ainur who died could indeed end up in the Void, and what we saw with Saruman and Sauron was in fact the immediate prelude to them ending up in the Void.