Former Tennessee Sen. William E. Brock III, backed by a diverse coalition of Republicans from all regions of the country, yesterday was elected national chairman of the GOP.

The new chairman takes over at a time when the Republican Party is at its lowest ebb since the Depression after losing the White House and having its minority in Congress and the statehouses further diminished. Though Brock spoke of the "incredible opportunity" facing the GOP, he also warned that the party will sink even lower unless it is able to capture legislatures during the next four years and influence congressional reapportionment.

Brock, whose reputation as a political organizer overcame the stigma of being a recently defeated one-term senator, was chosen after three secret ballots of the Republican National Committee. He defeated Utah State GOP Chairman Richard Richards, the runnerup on all three ballots, and three others.

Brock's election was a demonstration of independence by the national committee and in some respects a setback for the country's two most prominent Republicans, President Ford and Ronald Reagan.

Ford's effort to make his former campaign chairman, James A. Baker III of Texas, the national chairman collapsed this week when Baker proved unwilling to battle for the job and withdrew. Reagan supported Richards, and this helped prevent the Utah chairman from attracting moderate support.

Maryland, Rebublican Charman David R.Forward, a Brook supporter, said that one of his candidate's chief attractions was that this election enabled the party to avoid any further Ford-Reagan conflict.

Brock led Richards narrowly on the first ballot yesterday. He had 54 votes to 48 for Richards, with 82 needed to win. The other votes were divided among White House deputy assistant Arthur A. Fletcher, 22: Ohio GOP Chairman Kent B. McGough, 20 and District of Columbia national committeeman Robert S. Carter 16.

Two other prospective candidates never made it to the balloting. Indiana GOP Chairman Thomas S. Milligan withdrew when it became clear he lacked unified midwestern support, and Buehl Berentson, a former executive director of the Republican Governors Association, was unable to find a sufficient number of nominators.

On the second ballot the three trailing candidates lost ground, with most of their former support going to Brock. After that ballot. Brock led with 70 votes to 48 for Richards.

The national committee proceeded immediately to a third ballot in the packed ballroom of the Washington Hilton, and by this time it was apparent that Brock would win. His supporters kept the tally as the votes were called out on the rostrum and broke out with a roar of applause when the 82d was reached. The final tally, before a motion was accepted to make the vote unanimous, showed Brock with 90, Richards 46, McGough 14, Carter 6, and Fletcher 5, with one abstention.

One of the surprises of the balloting was the relatively strong first-ballot showing of Fletcher, believed to be the first black ever nominated for Republican national chairman.

Brock pledged after his election to embark on a strong campaign to attract minority voters, women and young people, and he said he would ask Fletcher to play "a tangible role."

"This party cannot just open its doors," Brock said in an acceptance speech prepared before the balloting. "It has got to go out and bring people in, and in doing so, give them a real voice in our leadership and in the development of our objectives. That means stirring the waters."

The new chairman takes over at a time when the Republican Party is at its lowest ebb since the Depression after losing the White House and having its minority in Congress and the statehouse further diminished. Though Brock spoke of the "incredible opportunity" facing the GOP he also warned that the party will sink even lower unless it is able to capture legislatures during the next four years and influence congressional reapportionment.

Brock 46, has shown some willingness to depart from political orthodoxy in the past, notably when he shocked his Republican conservatives in the Senate by supporting a postcard registration bill favored by the Democrats.

During his six years in the Senate Brock won a reputation as a highly competent political organizer who involved young people in party activities. He was upset in last November's election by Democrat James R. Sasser, who exploited the unwillingness of Brock, a millionaire heir to a candy fortune, to reveal the contents of a blind trust he establised when he went to the Senate in 1971.

Brock's defeat was his chief obstacle when he started out to campaign for the GOP chairmanship last November soon after Mary Louise Smith announced her resignation.

But his stock rose steadily when the usually infuential midwestern power brokers of the GOP national committee were unable to agree upon a candidate and when former Texas Gov. John B. Connally said he was unwilling to he a full time chairman, as party rules require. This collapsed a growing Southern move for Connally and made Brock the favorite in this region.