[Ivriniel]

....and

...cleaving the undead flesh, breaking the spell that knit his unseen sinews to his will.

This seems to me a significant moment in the narrative commentary about the 'what' of a Nazgul. Something Sauron 'did' via the Rings --a spell-- that somehow --knit-- that vestige of formerly mortal flesh --to will of might of mind--. I wonder now: Tolkien often spoke of Mind holding the Body of Men to Life. Numenoreans could yield their life at old age--with and act of will, or cling until death ended their body. This 'will' facility was explicitly noted in other races, especially the Elves and Elven Mind fortifying body through 'the other world' (Glorfindel being in 'two worlds' at the confrontation of the Ringwraiths).

This implies Sauronic Mind/Will forged connection or extension of Maia Mind to form the variation on flesh that was a Nazgul. Sauronic Mind also destroyed "feasted" on mortal living flesh as the process of transformation occurred.

On another train of thought, the formidable size of the Witch King was noted by Tolkien. Mass this seemed to imply. The flesh was unseen by eye of sight of mortal mind. It could be seen by Elf and by eye of sight of mortal wearing The One. The reason for the invisibility, I would assume, was the same as implied by what Tolkien wrote of Frodo in Elrond's healing room. Some 'fading' of flesh after the Ringwraith attack. Elrond didn't put Frodo completely together again. Some transformation of his flesh had occurred.

The Rings of Power (the Nine made them Invisible when used) is what seems a reasonable conclusion to draw. And each use removes a little bit more of the mortal fibre as the Mind of the User strengthens in synchronisation with Sauraon's Might of Mind. Then at a critical threshold, Sauron rocks in very closely and blasts away the remaining capacity to reverse the process and enslaves.