TOM HARKIN, methodically gathering Democratic support, appears to be taking a narrow lead over his rivals for the party's 1992 presidential nomination.A survey of five early-voting states - Iowa, South Dakota, Minnesota, New Hampshire, and Maine - finds Mr. Harkin is organizing quicker, gathering more endorsements, and exciting larger crowds than his five major Democratic opponents. The Iowa senator's current advantage, however, could quickly evaporate if New York Gov. Mario Cuomo enters the race later this month, as some analysts expect. Governor Cuomo would particularly threaten Harkin's support from organized labor, a key source of the senator's strength. Harkin, who breathes partisan fire at President Bush, seems to be particularly popular with liberal activists and with unionists worried about the loss of American jobs. At an AFL-CIO convention this week in Detroit, he told an audience with his usual Trumanesque flourish: "When I'm president of the United States, every double-breasting, scab-hiring, union-busting employer in America will know that the working people of America have a friend in the White House." Double-breasting refers to companies that have both union and nonunion shops, sometimes as a way of breaking the union. Harkin entered the race with at least one built-in advantage - the fact that he comes from Iowa, scene of the first-in-the-nation presidential caucuses on Feb. 10. Hugh Winebrenner, a political scientist at Drake University in Des Moines, Iowa, says there is no doubt Harkin will win handily in his home state. That has forced Harkin's opponents to look elsewhere for early support. But in several other crucial states, sources say that Harkin has either moved out front, or is challenging the leader. Those states include: Minnesota. Harkin, by getting in early, has won the support of key Minnesota politicians, including as many as 50 of the state's 120 Democratic legislators. "Harkin has the most aggressive campaign organization and has worked the hardest," says state party chairman Todd Otis. New Hampshire. Harkin is rapidly organizing the Granite State with the help of labor unions. Former US Sen. Paul Tsongas, from next-door Massachusetts, probably has the best team in the state, and has conducted extensive door-to-door canvassing. But informed Democratic sources say that Harkin is "catching up rapidly." South Dakota. This early primary state could be a battleground between Harkin and his Midwest rival for the nomination, Sen. Bob Kerrey of Nebraska. But by acting swiftly, Harkin has jumped into a significant lead. Maine. Several other candidates, including Arkansas Gov. Bill Clinton, Mr. Tsongas, and Senator Kerrey, got into this state sooner than Harkin. But with union support, Harkin is expected to be among the top two finishers, according to Maine analysts. In every state, however, insiders warn that if Governor Cuomo enters, the picture will change dramatically. A recent national poll by the Times Mirror Center for the People & the Press found Cuomo far more popular than Harkin or any other Democrat. The key for Harkin will be momentum. In recent years, presidential campaigners have attempted to build momentum, first, by doing well in Iowa, then, by winning in New Hampshire, and following that by running strongly in early states like South Dakota and Minnesota. The early winners get big media coverage, more money from contributors, and enthusiastic volunteers. Harkin's Iowa strength makes New Hampshire, the second state to vote, even more important than usual. In New Hampshire, Harkin appears to have built the second-best organization after Tsongas, who can rely on nearby Massachusetts volunteers. New Hampshire sources express a word of caution about Harkin's prospects there, however. They say that Tsongas, who is often casually dismissed by political pundits, appears to be well positioned in New Hampshire. They note that only about 100,000 people will vote in the primary, and that Tsongas may very well be able to win 25,000 or 30,000 of them, based on his organizational effort. That would probably be enough votes to place him either first or second. Yet what has most impressed some analysts is Harkin's swiftness in organizing support in third-round states like South Dakota and Minnesota, where he could follow-up a first-place or second-place finish in New Hampshire. By the time the third round of voting begins, some analysts think, it could be too late for a number of the other contenders, including Governor Clinton, Gov. L. Douglas Wilder of Virginia, former Gov. Jerry Brown of California, and perhaps Kerrey. Steve Hildebrand, co-director of the South Dakota Democratic Party, says Harkin already has coordinators working in 90 percent of his state's counties. "Kerrey doesn't have anything like that yet," Mr. Hildebrand says, and other candidates have almost no presence there. One Democratic official, speaking privately, says Harkin may be even stronger in Minnesota than in South Dakota. "I think he's got Minnesota wrapped up," the official says. "He has a phenomenal organization there.... And there's hardly an elected official in Minnesota who hasn't endorsed Tom Harkin."