It is settled pretty strongly that life is given by Illuvatar and the Gift Of Illuvatar to Men is death. And yet, we see the Nazgul (and the Dead Men of Dunharrow) refusing the gift and have some sort of existence for an extremely long time. Is there a canonical answer on how could this happen?
It is clear that the Ainur have the power to delay the Gift of Men. There are numerous examples. After Beren's death, his soul resides long enough in Mandos for Luthien to make a plea on his behalf, and the pair are restored to life. Their descendants are allowed to choose whether they will take the Gift or remain tied to Arda as elves.
Sauron, as a much lesser being than Manwe, probably lacks to power to change whether a given being is man or elf. (Luthien, most notably, married into humanity.) However, he does have the power to extend the lifetimes of men far beyond their normal span. For the Nazgul, their lives seem to be extended essentially indefinitely, although it is an open question whether they would still have died eventually, had their rings not lost their power. But the Ringwraiths are still men, and with the destruction of the ring, they died and eventually passed beyond the circles of the world. In one of his letters, Tolkien suggested that Sauron could have recreated the Lord of the Nazgul after he was slain, but that was only possible while the power of the ring lasted.
The Dead Men of Dunharrow were cursed to remain by an even lesser power, a king of the Dunadain, but Isildur apparently had enough ability to delay the departure of the oathbreakers as well. When their oath was fulfilled with the capture of the corsairs' ships, they too departed from Arda.
There are other possible examples as well--the barrow wights, for instance. What is clear is that magical power can be used to delay the Gift of Men. This can mean keeping men alive beyond the normal span of their years (as with the Nazgul, or even the long-lived Numenoreans). It can also mean that the souls of the departed can be prevented from leaving Arda immediately after death, although not forever. (This happened to Beren, the Dead Men, and possibly also the Nazgul.)
The Nazgul and Men of Dunharrow didn't really refuse Illuvatar's gift of death rather they were cursed and didn't have a choice.
The Silmarillion goes into detail of The Nazgul's predicament. Men proved easier to ensnare.Those who used the Nine Rings became mighty in their day: kings, sorcerers, and warriors of old. They obtained glory and great wealth, yet it turned to their undoing. They had what seemed to be unending life, yet life became unbearable to them. They could walk, if they would, unseen by all eyes in this world beneath the sun, and they could see things in worlds invisible to mortal men, but too often they beheld only the phantoms and delusions of Sauron. And one by one, they fell under the thralldom of the ring that they bore and under the domination of the One Ring, which was Sauron's.
Even without outright supernatural intervention, there seems to be some 'flexibility' built in -- Hobbits are essentially Men, and they live longer: the Old Took made it to 150. So the Gift of Men doesn't seem to demand an absolute upper limit of circa 120 as we see in real life.
And the Valar can't undo the gifts of Iluvatar, but they could make the Numenoreans live to 200-plus.
The Ringwraiths (and Gollum) are much older than that, though. And the life-prolonging effect of the Rings seems to be somewhat different. Aragorn doesn't seem to have lost anything by his extended lifespan, but the Nazgul have literally faded away, their bodies seem to have lost most of their 'solidity' - Gandalf says
the black robes are real robes that they wear to give shape to their nothingness when they have dealings with the living
Gollum is pretty twisted by the Ring, and even Bilbo (who had the Ring for far less time) describes the effect as:
"sort of stretched, if you know what I mean: like butter that has been scraped over too much bread."
It seems the Rings 'cheat' by increasing 'quantity' of life without increasing 'quality'.