When the Nazgûl lose their bodily form, what happens to their rings?

[Boromir88]

Sauron possessed the 9 rings, as some of the 7, as you correctly recall.

Unfinished Tales:

...mightiest servants, the Ring-wraiths, who had no will but his own, being each utterly subservient to the ring that had enslaved him, which Sauron held

...were entirely enslaved to their Nine Rings, which [Sauron] now himself held

And in LOTR:

...the Nine [Sauron] has gathered to himself; the Seven also, or else they are destroyed

You saw the Eye of him that holds the Seven and the Nine

My thoughts as to why he had them, is what Tolkien wrote in Letter 246 (another place where I believe Tolkien mentions Sauron held the 9 rings). And that is Sauron had a way to control his Nazgul, even without the One Ring. In Letter 246, Tolkien sets up a hypothetical scenario about Frodo the "ring-lord" claiming the One Ring, in the Sammath Naur, but the Nazgul would still take Frodo from Mount Doom and bring him to Sauron. As it was their 9 rings that that Nazgul were enthralled to, not the One.