A celebrity-studded line of charitable contributors -- overflowing the route in Washington, which will wind through the White House grounds to include President Reagan and his family, but sparse in the Southwest desert -- will hold Hands Across America tomorrow, singing songs and urging America to remember its own hungry and homeless.

The 3 p.m. handholding extravaganza is being billed as "the biggest charity event ever," and organizers say they have been so swamped at the last minute with people who want to join the line that they don't have an accurate count on how many participants to expect.

"There's just no way to judge the response now . . . it's hot," said Fred Droz, the veteran political organizer who is national director of the Hands Across America project. While conceding that there will be gaps in the 4,100-mile coast-to-coast human chain, Droz said the event could end up raising more than $50 million to help shelter and feed the nation's poor.

"The reality is that we will have several million people doing something that day together at one moment," he said.

President Reagan was among the late-comers, abruptly changing plans yesterday and deciding to join the line. Presidential spokesman Larry Speakes said Reagan and his wife Nancy decided to join the project after a family discussion Thursday night.

Speakes quoted Reagan as saying, "This is a uniquely American way to help our fellow man. The Reagan family will do its part." Speakes denied that the decision grew out of controversy prompted by the president's remark to a high school group this week that hunger in America is due not to a denial of aid to anyone but to some people's ignorance of where to get food.

Speakes said the Reagans will join the line outside the north door of the executive mansion and are inviting members and families of White House and Secret Service staffs to join them.

In Washington and elsewhere, late-comers are being advised to "just show up" where they want on the route, although organizers will broadcast appeals for people to fill specific gaps in the line. Participants who have not yet purchased places in the line will be given postage-paid pledge envelopes for mailing in their contributions. There will be no cash collected along the route.

The attempt to link New York's Battery Park with Long Beach, Calif., is no small undertaking. It means rounding up at least 5.4 million people and deploying them across 16 states, 556 cities and towns, 10 rivers, three mountain ranges and two deserts. Put another way: If the Hands Across America participants tried to pass a three-second message along the route, it would take more than six months.

Tony Tortorici, director of public relations for Coca-Cola USA, a major sponsor, said the firm decided to help bankroll the project last October after pop star manager Ken Kragen, of USA for Africa fame, went to the company with the idea.

"We felt the mood of America was such that Americans were eager to help Americans, and Coca-Cola is an integral part of American life," Tortorici said. "It's a chance for every American to be a celebrity for 15 minutes and to make a wonderfully positive statement about helping others."

Except for the Sousa Bridge and the emergency room entrance to George Washington University Hospital, Hands organizers in the District hope to fill every segment of the 29-mile route through the city and are making plans to bus overflow crowds at Dupont Circle to gaps along Massachusetts Avenue NW. The full line will block 243 intersections here, with police stopping traffic for about 10 minutes just before 3 p.m.

In Maryland, the 156-mile Hands Across America route has been harder to fill, especially in the rural, western part of the state. Organizers threatened Thursday to use cows, if necessary, to close gaps in the line.

"There's a lot of ground to cover," said one statewide project official.

To assist in crowd control in the more urban areas, the Hands Across America organizers plan to set up staging areas along the route to provide entertainment, beginning at 1 p.m., and support services. There will be 21 such gathering places in the District, each with police officers, crowd monitors, toilets, water, first aid facilities and celebrity emcees and entertainers.

All activities related to Hands Across America are supposed to end before 3:30 p.m., according to organizers.

The amount of grass-roots support for the project -- schools, unions, civic associations, churches, clubs -- is impressive in its variety. But it is corporate America, specifically about 700 companies, that provided the lion's share of seed money and other assistance for the event.

Project organizers, following the example of the 1984 Olympics, said they tapped corporate resources by appealing to the increased desire of companies to be thought of as socially responsible. It is that very visible corporate involvement, however, that is generating a slightly negative backlash about Hands Across America among some people who resent the commercialization of the event.

Many existing charitable groups, while supporting the project, worry that national attention for one day does not promise long-term solutions to the problems of the hungry and homeless. Others argue that corporations should have been doing more for the poor a lot sooner.

"We could not have done this project without the funding, and there's nothing wrong if a corporation gets the benefit" of publicity, said Carolyn Anderson, national sponsor coordinator for Hands Across America