Celebrating a primary campaign that had a worrisome start in Iowa, George Bush at last could lay full claim Thursday to his party's White House nomination with the backing of his boss, President Reagan.

Now the hard part begins.Even as Republicans were toasting their recent past under Reagan and their hoped-for future under Bush, Democrats Michael Dukakis and Jesse Jackson kept campaigning Wednesday for the right to oppose the vice president in November.

Dukakis, who expects to win his party's nomination, was in Washington Thursday for a political pilgrimage to Capitol Hill. Jackson was on the West Coast and Bush, upbeat from the outpouring of partisan support at a fund-raising dinner Wednesday night, headed back to California to rejoin the campaign.

It was a well-orchestrated chain of events Wednesday in which Bush became the lone survivor of the battle for the GOP nomination and pick up the formal, if tepid, endorsement of the most popular president in years.

At a midmorning White House meeting, ex-television evangelist Pat Robertson announced he no longer would be the last major Republican competing with Bush and was "wholeheartedly endorsing" the vice president.

"We're going to work harmoniously and in unity together," said Robertson, who stunned Bush just three months ago by finishing second to Senate Republican leader Robert Dole of Kansas in the Iowa presidential caucuses.

Bush, of course, fought back from that defeat - and Wednesday night, as almost 4,000 well-connected Republicans dined on roast filet of beef at a fund-raiser that garnered more than $5 million for congressional campaign coffers, he received what a campaign aide called "the jewel in the crown."

Reagan mentioned Bush only three times in the course of an endorsement speech that exceeded 20 minutes and even mispronounced his name at the end. But the event - hardly a surprise - was exploited for maximum media exposure.

"I intend to campaign as hard as I can," Reagan declared in breaking a self-imposed silence on the campaign. "My candidate is a former member of Congress, ambassador to China, director of the CIA and national chairman of the Republican Party. I'm going to work as hard as I can to make George Bush the next president of the United States."

On cue, the crowd roared, balloons dropped from the rafters and the Bushes joined the Reagans on stage, their hands clasped and arms raised in a scene that could preface August's Republican National Convention in New Orleans.

The president then stepped back to the microphone and crowed, "Now it's on to New Orleans and on to the White House!"

Despite the grand finale, a frequently restrained atmosphere Wednesday night signaled possible problems ahead for Bush, who in the months ahead will try to galvanize the Reagan constituency of 1980 and 1984 without the same personal and political qualities.

Moreover, though sources said Reagan wrote his own remarks, the endorsement fell far short of the praise some reporters had been led to believe he would heap on a man being rewarded for years of steadfast political allegiance.