I think you can even pinpoint its origin: January 26, 1911, the premiere of Richard Strauss's Der Rosenkavalier, the most epic wistful au revoir in the repertory (predating, not incidentally, both Stravinsky's Rite and Schoenberg's Pierrot). In stylistic retreat from the dissonant expressionism of Salome and Elektra, Strauss let his Marschallin bid adieu to her Octavian, her youth, aristocratic Vienna, what have you, with a trio so languorous Strauss would have let it run on into 1912 if he thought he could get away with it. Late Romanticism had already produced its share of goodbyes—Mahler's Das Lied von der Erde, with its own so-long farewell "Der Abschied," was also premiered in 1911—but I think it was Rosenkavalier that made lush, chromatic tonality a signal of nostalgic intent. Ravel had it stand in for lost pre-WWI Europe in La Valse, Rachmaninoff made a cottage industry lacing his own rich potions with "Dies irae" reminders, Strauss himself followed up Rosenkavalier with another epic trio of send-offs in the 1940s—Capriccio, Metamorphosen, and the Four Last Songs. (The style would breed a second round of nostalgia in the 1970s and 80s via film scores, as John Williams and the like resurrected the late-Romantic, Golden-Age Hollywood style of Korngold, Steiner, &c.)
Nostalgia was largely a creation of Romanticism in the first place, with music playing a crucial role. It was the ranz des vaches (literally, the call to the cows), the song of the Alpine herdsman, that originally gave rise to the whole notion of nostalgia, often referred to as the mal du Suisse in the early days—according to Rousseau, the ranz des vaches
was so generally beloved among the Swiss, that it was forbidden to be play'd in their troops under pain of death, because it made them burst into tears, desert, or die, whoever heard it; so great a desire did it excite in them of returning to their country.The increasing 19th-century awareness of history as a historical force—most fully articulated in Marx's historiography or Nietzsche's post-French-Revolution "sixth sense"—would transfer that homesickness to eras, timesickness as it were. So it makes sense that late Romantic music would adopt nostalgia as its own, being the first style to run its course with nostalgia imprinted on our collective consciousness.
But I always find it interesting that it's late Romanticism that got the nod. That semiotically nostalgic cast had become more potent with the rise of atonality—when George Rochberg looked to rewind atonal modernism, for example, he opted for Mahlerian tonality—but, of course, the seeds of atonality can be found in the adventurous chromaticism of late Romanticism. And, oddly enough, it proved an effective survival strategy. Strauss, Mahler, Debussy, and the like are regarded as far more effective box-office draws in classical music than their immediate, direct atonal descendants, and those contemporary composers with the most mainstream cultural traction (I'm looking at you, John Adams) are those whose vocabularies most extensively borrow from the twilight of the Romantics. To paraphrase another grandiose cultural artifact, audiences apparently hate to see tonality go, but they love to watch it leave.