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

[Pitchwife]

Actually, the idea that Sauron held the Nine Rings seems to have been the original one, going back to the early drafts.
From The Return of the Shadow (HoME VI), Of Gollum and the Ring:

[...] the men-wraiths are servants of the Lord, and they brought all their rings back to him; till at last he gathered all into his hands again that had not been destroyed by fire[...]

And from RS, Ancient History:

But all the Nine Rings of Men have gone back to Sauron, and borne with them their possessors, kings, warriors, and wizards of old, who became Ringwraiths and served the maker, and were his most terrible servants.

Curiously, Gandalf is the speaker in both cases; so what possessed him to state "The Nine the Nazgûl keep" at the Council of Elrond? Maybe just a temporary lapse of memory by Tolkien, the original idea resurfacing later in UT and Letters?

Boro:

As it was their 9 rings that that Nazgul were enthralled to, not the One.

Saruman, expert in ring-lore, thought otherwise (UT, The Hunt for the Ring):

I know what you seek, though you do not name it. I have it not, as surely its servants perceive without telling; for if I had it, then you would bow before me and call me Lord.[my emphasis, Pw.]

He may have been mistaken, of course.

TGEW:

What I mean is, the Wraiths would always want to come back to their rings in hopes that Sauron, being the nice, generous guy that he is, would give them back.

I quite like this thought. Maybe in this respect the Nazgûl weren't that much different from Gollum - addicted to their rings, and driven mad by more than an age of withdrawal?
"Please masster... give preciouss back to good Khamûl!"