Republicans, the women in long fur coats, circled the lobby of the L'Enfant Plaza Hotel yesterday, waiting to go off to parties. There was excited chatter, an air of euphoria.A big sign in the hotel said, "America, a Great New Beginning."

Off the back of the hotel in the shadows of the long inaugural weekend, the nation's big-city mayors were winding up a three-day meeting in a mood of gloom and fear that the new national administration will cut deep into their social programs.

For the most part they tried to put on a brave face, but shouts and some screaming came from one of the U.S. Conference of Mayor's old tigers, Henry Maier, the 20-year mayor of Milwaukee.

"I say to you that if you say [cities] can live without federal help, God bless you," the fullthroated Maier said, pounding the table. "Some of us are on the brink of catastrophe. mYou don't believe me, go look at those bond ratings. . . . We have city after city on the verge of collapse.

"Now we're being told as we are about to be booted down the hill further that we ought to allow them to boot us -- take it in good grace -- and I have only one thing to say to you. To hell with that!"

Though Maier was outspoken, pained litanies came quietly from other Frost Belt mayors, particularly Coleman Young of Detroit and Doug DeGood of Toledo, whose cities are suffering from the effects of sharp slides in the auto and steel industries.

Uncertainty was all around the mayor's conference: uncertainty about what programs President Reagan would cut; uncertainty about who would fill the important sub-Cabinet jobs; uncertainty about what kind of relationships the mayors, the great number of them liberal Democrats, would be able to forge with the conservative Republicans newly in charge.

Three designated Cabinet secretaries -- Drew Lewis of Transportation, James Edwards of Energy and Samuel Pierce of Housing and Urban Development -- spoke to them, imparting about as much information about strategy as the Oakland Raiders are willing to give the Philadelphia Eagles as they head into the Super Bowl.

When San Fransisco Mayor Dianne Feinstein asked Pierce if the mayors would have access to him, the HUD secretary-designate said stiffly: "No questions."

Was that shyness or arrogance, the mayors later debated among themselves.

The new Republican leaders on Capitol Hill who spoke to them were more forthcoming.

Sen. David Durenberger (R-Minn.), chairman of the subcommittee on intergovernmental relations, declared this a time for change.

"We have reached the limit [on income security programs]," he declared. "More than that, it is generally agreed that we have gone somewhat beyond the limit and retrenchment is necessary. In our weakened [economic] condition, we cannot afford to meet the weakest claims on the public treasury."

And he mentioned a few other harsh, though hardly new, facts of life: Whereas Democratic presidents tend to enjoy strong ties with mayors, Republicans tend to favor governors, he said. That brought on nervous questions.

The conference ended with the mayors distributing a "National Urban Investment Program" whose wish list of new programs would cost a billion dollars, according to conference President Richard G. Hatcher of Gary, Ind.

Yet, Hatcher said, he and the other mayors wanted, like the new administration, to save money, increase effectiveness and reduce waste and abuse.

How, he was asked. Which of the array of federal programs are essential, which can be cut without great harm?

Hatcher and Mayor Lee Alexander of Syracuse, N.Y., ducked the question.

Feinstein of San Fransisco said: "I think it's unfair to put the mayors in this position until we see the priorities in the president's budget next month. My own view is that we've got to take a look at Social Security and the increases."

The question triggered another shouting sermon from Maier of Milwaukee.

"I think the most ignorant damn people in America are the media because you're all middle-class," he declared. "Most of you come from middle-class homes, and you really don't get the smell of these central cities and the problems of these central cities. If you did, your number one priority, like ours, would be to make damn sure our funding isn't cut back, because if it is cut back, the domestic tranquility -- I promise you -- is going to be very seriously disturbed."