This is from the Boston ATLAS 7 June 1844.
By mid-May 1844, two weeks before the Baltimore Convention, Amy S. Greenberg tells us, (25) Polk "had long been known as 'Young Hickory."
Now, I would take Herman Melville's mother's word for a thousand pound[s], but I am more cautious about what his brother Allan says. However, Allan does say to him in his October 1844 welcome-home letter that "the honor of christening Polk 'Young Hickory'" belonged to their older brother, Gansevoort Melville.
That is, Allan thought that in his great City Hall Park speech of 4 June 1844 Gansevoort Melville had coined the nickname.
The 6 June 1844 Albany ARGUS gives more of Gansevoort's elaboration:
. . . [W]e will re-Christen him [Polk]. Hereafter he shall be known by the name that we now give him--it is, Young Hickory. [Here the cheering was deafening, and continued for some moments. A voice: "You're a good twig of Old Hickory, too!"--laughter and renewed cheering.] We have had one old hickory tree. Its trunk is yet green and undecayed Sixteen millions of Americans have reposed under its shade in peace and happiness. It is yet vigorous--but it cannot live for ever. And now to take its place is springing up at its very side, a tall and noble sapling. . . .We and our children will yet live in prosperity under the broad branches of this one young Hickory tree. On the 4th day of March next, that young Hickory will be transplanted by the People, in the People's House at Washington, and you, and I, and all of us, will assist in that transplanting . . . .
Now, I see that Isaiah Wright may have used the phrase in Boston on 4 June 1844 but I don't see it any earlier. Do you?