It was no coincidence that leaders of two huge public employe unions chose yesterday to spring on a waiting nation their analysis of the impact of cuts in federal aid to state and local governments.

The cuts, they said in a news release, have brought state and local governments "near the brink of disaster" and further trims would "push them over the edge." The release did not specifically mention that many thousands of public employe jobs would be affected; it just said that many services for needy people had been eliminated.

While the results of the study have been known to the American Federation of State, County and Municipal Employes and the Public Employe Department of the AFL-CIO for some days now, the news conference was held the day before President Reagan is to give his State of the Union message.

What better moment, AFL-CIO spokesman Tom Fahey said, for the unions to complain about their state? Such timing is "hardly original on our part," he said.

Indeed. With the president's fiscal 1984 budget soon to follow, this has become one of the most popular periods of the year for various interest groups to come to Washington, call news conferences and either decry what they think is coming or suggest the appropriate course for the administration and Congress.

In the past two weeks, groups as varied as the Heritage Foundation, the National Urban League, the House Republican Research Committee, the U.S. Conference of Mayors, the National Governors' Association, the National Center for Legislative Research and the Sheet Metal Workers Union have been busy peddling agendas.

Sometimes it works, said Fahey. "Last July, just before the 1982 tax cut went into effect, we released a study demonstrating that unless you make big-figure bucks, the decrease amounted to an increase." The release was timed to coincide with the Democratic National Committee's mini-convention, he said, and thus received broad coverage.

The Heritage Foundation has also proved adept at timing. The group, generally labeled as conservative, published a document in 1980 that laid out an agenda for Reagan, and much of it found its way into the president's program.

On Friday, Heritage released a new book, "Agenda '83," which suggests a course for Reagan at midterm. Chapters were leaked to different news organizations to guarantee good coverage on subjects from defense to deregulation, and then a big news conference was held to discuss the entire document.

The press wasn't the only target. Weeks ago, according to Richard N. Holwill, vice president at Heritage, the chapters of "Agenda '83" were sent to administration officials, including number crunchers at the Office of Management and Budget.

"If you want people to pay attention to you, you've got to get to them before they make up their minds," Holwill said. " . . . Accomplishing 100 percent of what is sought may not be do-able, but if it is not, it is still terribly important we provide intellectual leadership to get people talking about things."

The Conference of Mayors is meeting here this week. Executive Director John J. Gunther said that "we started our first organized meeting with Mr. Hoover right after the election of 1932. We didn't get very far . . . . "

There have been successes, however. "In 1981," Gunther said, "we found out from Reagan and his staff they were going to get rid of UDAG the Urban Development Action Grant, federal seed money for private urban development . At that meeting they raised all kinds of questions" and the program was saved.

Not all messages are delivered in January. For 14 years the Brookings Institution has been producing a document titled "Setting National Priorities," which is an analysis of the president's budget proposal, and it comes out in May or June.

Joseph A. Pechman, director of economic study at Brookings who heads that effort, said that, obviously, the budget must be there to analyze before Brookings can analyze it. "We don't set up a counter budget," he said. "In fact, in many instances, we agree with a particular administration.

" . . . If Brookings weren't at least providing information that would help people make up their minds about policy, I wouldn't be at Brookings," he said. "I think that's the point of being a social science research worker."