When Vice President Bush went to the Gridiron Club dinner here last night, on one of his first free evenings since the fast-paced, four-continent, 14-day diplomatic journey that consumed most of his March calendar, he heard himself serenaded to the tune, "A Bird in a Gilded Cage."

A bird, maybe. Caged, definitely not.

The 28,164-mile sprint, which took Bush from the sandstorms of sub-Sarahan Africa to the chill of Moscow and then to the heat of Grenada, Honduras and Brazil, tested his mental agility as much as his physical stamina.

By the accounts of administration officials and journalists who accompanied him on parts of the marathon, he acquitted himself well on both scores, but it took a toll.

Sitting in his White House office on Thursday, between Republican fund-raising dinner assignments in Philadelphia and Orlando, the vice president, 60, called this month's trip "the most demanding" of any of the diplomatic missions he has undertaken for President Reagan.

It also may have been one of the more rewarding -- politically.

On the African leg of his journey, which included visits to refugee camps in Sudan, he was accompanied by the Rev. Pat Robertson and joined by the Rev. Jerry Falwell. Ostensibly, the two television evangelists were there because of their sponsorship of major feeding programs for the famine victims, but their presence also symbolized a growing rapport between Bush and leaders of the New Right Christian groups that have played an increasingly important role in Republican presidential politics.

His status as the first U.S. official to confer with Mikhail Gorbachev, the new leader of the Soviet Union, put him on the "Today" show and in hundreds of newspaper stories. As he travels the Republican banquet circuit, he can salt his speeches with impressions of the Kremlin boss that no other Republican politician can duplicate.

Most important, by carrying out yet another assignment from the president, without mishap, Bush solidified his status as Reagan's implicit choice as a successor.

When the White House on Thursday allowed photographers to snap Reagan and Bush sitting down for their weekly private lunch -- a photo that turned up on Page 1 of Friday's New York Times -- it was one more sign to the politically attuned that, no matter whether he ever endorses Bush formally, Reagan is more than willing to let his vice president use the White House as the launching pad for his 1988 candidacy.

Bush said in the interview that he hasn't put the trip and its byproducts "into an '88 context. I put it into just doing my job. But if it's an asset, and people think that, so much the better."

It is an asset for which Bush pays a price, because on Thursday, as he headed for a brief Florida rest sandwiched between fund-raisers, fatigue lines were etched deep around his eyes.

By the log his staff keeps, Bush flew for 57 hours between his March 3 departure and his March 17 return, including one mind-bending 18-hour hop from Moscow to Grenada with two refueling stops, eight hours of ceremonies on the island, then a flight to Brazil.

Almost as tough as the time-zone and temperature changes were the changes in political climate.

In Moscow, after the funeral ceremonies for Konstantin Chernenko, Bush's job was to test the prospects for detente with the new Soviet leader. In Grenada, where he stood on a soccer field in an emotional crowd that was waving anticommunist signs, his task was to reconcile local leaders to the withdrawal of the last of the U.S. troops that stormed ashore late in 1983 to drive off the Cubans and the Marxist government they supported.

The toughest part of the trip emotionally came at the very start, in the drought-stricken refugee camps of Sudan. Bush said that holding "7-year-old kids who weigh less than my grandson, Sam, did at seven months . . . it just makes a profound impression on you."

Washington Post correspondent Jonathan C. Randal, who accompanied Bush to the refugee camps, said he seemed at times "stiff and ill at ease" and oblivious to the photographers and television cameramen seeking human-interest shots. To some in the entourage, Randal said, "it seemed an amateur-hour version of the much smoother trip by Sen. Edward M. Kennedy (D-Mass.) over much the same turf last Christmas."

But Bush had a political mission in Sudan, to prod President Jaafer Nimeri to make the economic and political reforms whose delay had held up $214 million in U.S. aid.

Using a carrot-and-stick approach, Bush announced release of $15 million in assistance, but reiterated that the rest would not be forthcoming until promised internal changes are made. His national security assistant, Donald P. Gregg, said private meetings with the country's first and second vice presidents -- representing feuding regions -- prompted Bush to press Nimeri publicly in his farewell statement for new moves aimed at political reconciliation.

After visits to Niger and Mali, Bush flew to Geneva to address a conference of countries supplying food aid to Africa, and from there was scheduled to return to the United States. Instead, he flew to Moscow to attend his third Soviet state funeral of the past four years.

The funerals are a sensitive subject with Bush. "What is overlooked by the Gridiron Club" in jokes about his role as the administration's designated mourner, he said, "is everything else that takes place there. You know, you have bilaterals with [Japanese Prime Minister Yasuhiro] Nakasone, with [West German Chancellor Helmut] Kohl, with [British Prime Minister] Margaret Thatcher, with [Indian Prime Minister] Rajiv Gandhi, with President [Mohammad] Zia [ul-Haq of Pakistan] -- there's five chief-of-state meetings within a period of less than 24 hours. I think you'd fly halfway around the world to do that anytime, if you're trying to make a contribution to foreign policy. But that is overlooked . . . . "

Bush said that what Reagan values most are the impressions of individual personalities. "The president asked me to make the first contact with [French President] Francois Mitterrand, the first contact with Gorbachev. He wants my impressions of how they react to things, and I give those to him in confidence . . . . He's a good listener."

The unexpected detour to Moscow required the marathon flight to Grenada, to maintain the schedule that had Bush serve as the top U.S. representative at ceremonies on the island and, a day later, at the inauguration of Brazil's newly elected civilian government.

There, Bush won more television attention by coolly turning aside Nicaraguan President Daniel Ortega's efforts to bait him into debate on U.S. policy in Central America.

Ortega, Bush said, reminded him of a former Republican House member from Long Island, Seymour Halpern, who, he said, "always stationed himself right on the aisle" on ceremonial occasions in the House, "and he'd be in the pictures, looking over Nixon's shoulder or whoever it was. Ortega made Sy Halpern look like an amateur."

The last stop was Honduras, Nicaragua's nervous neighbor, and despite the travel fatigue Bush still was enthusiastic. "I loved being with Suazo Honduran President Roberto Suazo Cordova in his home town of La Paz," he exclaimed. "There was no put-on. Our meeting was in his sister's house. It was real grass-roots stuff. And they turned the whole town out -- kids, flags, it was just wonderful."

There were no bands to greet Bush when he got back to Washington last Sunday, and homecoming meant no slackening of the pace.

On Monday, Bush was thrown into the White House battle for congressional approval of the MX missile. He made a quick trip to Baltimore to give a speech outlining administration arguments for the weapon, aimed partly at national press coverage and partly at Sen. Charles McC. Mathias Jr. The Maryland Republican, who voted against the MX last year, voted for it this time.

On Tuesday, he joined the president in last-minute lobbying on Capitol Hill, and, according to Republican leadership aides, was helpful "in smoothing some of the ruffled feathers" among GOP senators irked by White House leaks that suggested political retaliation against MX opponents.

Then came the Pennsylvania and Florida fund-raising trips, part of the push to retain Republican control of the Senate next year.

And at the end of the week, what did he hear? That he's "Only a Bush in a Gilded Cage. A pitiful sight to see."