The answer depends on exactly what the Nameless Things are. Are they indigenous creatures from Middle-Earth, perhaps created when Eru sent the Flame to be the heart of Arda?
And I will send forth into the Void the Flame Imperishable, and it shall be at the heart of the World, and the World shall Be; and those of you that will may go down into it.' And suddenly the Ainur saw afar off a light, as it were a cloud with a living heart of flame; and they knew that this was no vision only, but that Ilúvatar had made a new thing: Eä, the World that Is.
(The Silmarillion)
One of the strongest Tom Bombadil theories is that he is some kind of spirit indigenous to the world, created when Eru sent the Flame into Ea and made things Be. The reason many people believe this is that Bombadil makes a similar claim about himself: he's the Eldest.
‘Don't you know my name yet? That's the only answer. Tell me, who are you, alone, yourself and nameless? But you are young and I am old. Eldest, that's what I am. Mark my words, my friends: Tom was here before the river and the trees; Tom remembers the first raindrop and the first acorn. He made paths before the Big People, and saw the little People arriving. He was here before the Kings and the graves and the Barrow-wights. When the Elves passed westward, Tom was here already, before the seas were bent. He knew the dark under the stars when it was fearless - before the Dark Lord came from Outside.’ (The Lord of the Rings, the Fellowship of the Ring)
Given the other claims (before the first rain-drop and the first acorn, before the Elves past westward), this is usually understood as meaning that Tom was there before the Dark Lord Morgoth. But Morgoth was there 'from the first.'
And in this work the chief part was taken by Manwë and Aulë and Ulmo; but Melkor too was there from the first, (The Silmarillion)
(There's a better quote about Morgoth being the first of the Ainur in Ea, but I can't remember where it is.)
If Bombadil can claim to be Eldest, based on this, then maybe so can the Nameless Things be older than Sauron in the same sense - and if the Nameless Things came into being at the creation of the world, or at least before the Ainur (or specifically Sauron) entered it, then they would in this sense be older any of the Ainur - the Ainur at this time do not have an age, because they exist outside of Time, in the Timeless Halls and the Void outside of Ea.
Then those of the Amur who desired it arose and entered into the World at the beginning of Time; and it was their task to achieve it, and by their labours to fulfil the vision which they had seen.
(The Silmarillion)
But if they are Ainur, then some of the Ainur are described as older and younger than others.
Námo the elder dwells in Mandos, which is westward in Valinor. [...] Irmo the younger is the master of visions and dreams. [...] The spouse of Oromë is Vána, the Ever-young; she is the younger sister of Yavanna. (The Silmarillion)
It's hard to conceptualize this given the above - them existing outside of Time, and all, nobody could have been created before, at the same time as, or after anyone else, but maybe it's simply some kind of 'spiritual' difference analogous to age, such as when we're told Manwe and Melkor are "brethren in the thought of Illuvatar". In this case, the Nameless Things could be 'older' than Sauron or other Ainur. It could also simply be that the relative age of the Ainur was accounted from the time they entered Ea - but in this case, it would still be no problem for Nameless Thing as Ainur to be older than Sauron. Even if Sauron had known them in the Timeless Halls, who knows how much they might have changed from the time they entered the world until he did?
