In "The Minister's Black Veil," who is the nurse at Hooper's deathbed?

The nurse at Hooper's deathbed is Elizabeth, his former fiancee. 

Not much mention of Elizabeth is made at this point in the story, other than to name her as the nurse. This may be because it is more important for Hooper's personal story arc for her to simply be present, rather than to really do much of anything. 

Hooper's choice to wear the mysterious veil, obscuring his face, comes as much of a surprise to Elizabeth as it does to everyone else. Like others, she asks Hooper to remove it for her, appealing to their personal relationship. This was probably meant to fully round out the depiction of Hooper's resolution, showing that no common or earthly temptation was enough to change his mind. However, Elizabeth comes to fear the veil as others do, and breaks off the engagement when she sees that the veil will always separate her from Hooper. Hooper finds it sad that a piece of cloth should have this effect.

Elizabeth's presence at his bedside as he dies, with her affection for him having "endured in secret" tells us that she never stopped caring for Hooper, and while the veil had a powerful effect on his life, some people still saw him for who he was and were not completely superficial about it. It also signifies that the veil had a permanent isolating effect, preventing Hooper from having things that would have been available to him whenever he wanted, had he taken off the veil.

Who nurses Mr. Hooper on his deathbed in "The Minister's Black Veil"?

In "The Minister's Black Veil," Nathaniel Hawthorne tells of a minister named Mr. Hooper, who shows up at a Sunday service with a semitransparent black veil draped over his face. He preaches on secret sin, and the veil somehow lends significance and gravity to his words. From that moment he appears everywhere with the veil and refuses to take it off in public. It is a source of great speculation, awe, and even dread, but no one has the courage to confront him about it except one person.

Mr. Hooper has a fiancée named Elizabeth. She attempts to persuade him to take off the veil and explain why he put it on. He replies, however, that he is "bound to wear it ever." He begs her not to desert him, and she asks him once again to lift the veil. When he refuses, she leaves. From the time that Elizabeth leaves him, Mr. Hooper lives in abject loneliness. He does, though, become "a very efficient clergyman," always ready to help those in need.

When Mr. Hooper is on his deathbed, many people are present, such as the physician who is attending him, deacons, church members, and another minister. The nurse at his deathbed is Elizabeth, his former fiancée. Hawthorne writes,

There was the nurse, no hired handmaiden of death, but one whose calm affection had endured thus long in secrecy, in solitude, amid the chill of age, and would not perish, even at the dying hour. Who, but Elizabeth!

We see, then, that though Elizabeth left Mr. Hooper when he would not remove his veil, she secretly retained her love for him. As he is dying, she returns to him to be with him in his final hours.