Were the Nazgûl undead?

Question

Various sources refer to the Nazgûl as being undead. What was Tolkien's view on this, would he have classified the Nazgûl as undead?

Answer 1

The Witch-king is described as undead...

In Return of the King, when Merry slashes at him (emphasis mine):

So passed the sword of the Barrow-downs, work of Westernesse. But glad would he have been to know its fate who wrought it slowly long ago in the North-kingdom when the Dúnedain were young, and chief among their foes was the dread realm of Angmar and its sorcerer king. No other blade, not though mightier hands had wielded it, would have dealt that foe a wound so bitter, cleaving the undead flesh, breaking the spell that knit his unseen sinews to his will.

^{Return of the King Book V Chapter 6: "The Battle of the Pelennor Fields"}

...but they're probably actually alive

This seems commensurate with Gandalf's description of "wraithification" in Fellowship, where he remarks that Wraiths aren't dead (emphasis mine):

'A mortal, Frodo, who keeps one of the Great Rings, does not die, but he does not grow or obtain more life, he merely continues, until at last every minute is a weariness. And if he often uses the Ring to make himself invisible, he fades: he becomes in the end invisible permanently, and walks in the twilight under the eye of the dark power that rules the Rings.

^{Fellowship of the Ring Book I Chapter 2: "The Shadow of the Past"}

But let's go deeper. In Tolkien's mythology, "death" for incarnates means one of two things; you're dead if:

1. Your body and spirit have been separated. This is clearly not true of the Nazgûl; they obviously have some kind of physicality, since they're able to wear clothes and wield swords and ride horses and such. Although their physical presence is greatly lessened, they clearly have form of some kind

2. Your spirit has passed beyond the World. This is the kind of "death" we're talking about when we say the Elves are immortal. This, clearly, is also not true of the Nazgûl; their spirits are still bound within the World, though the mechanics of how this is possible is unclear.

Also notable is that Tolkien elsewhere discusses a different sort of "wraithification", and specifically notes that those wraiths do not die:

As ages passed the dominance of their fëar ever increased, 'consuming' their bodies (as has been noted). The end of this process is their 'fading', as Men have called it; for the body becomes at last, as it were, a mere memory held by the fëa; and that end has already been achieved in many regions of Middle-earth, so that the Elves are indeed deathless and may not be destroyed or changed.

^{History of Middle-earth X Morgoth's Ring Part 2 "The Second Phase" Chapter 3: "Laws and Customs Among the Eldar" Of Death and the Severance of Fëa and Hröndo [> Hröa]}

Answer 2

The term 'undead', for most people reading this site, refers to a specific fictional point of reference: the Vampire and friends as represented in pop culture and (more or less recent) fantasy. Tolkien's world does not use this category. When the text refers to the 'undead flesh', 'undead' is not a specific reference to a specific condition, but rather plain old descriptive, metaphorical, writing. It tells us that the witch-king isn't exactly normally alive, and is certainly not normally dead. It does not tell us that Middle-Earth includes a D&D-like class of creatures called 'undead' with a crisp definition.