They sang "Rule, Britannia" on Downing Street and threw rocks outside the Casa Rosada in Buenos Aires when they heard the news: The 10-week war in the Falkland Islands was over.

But here in Washington, spokesmen at the British and Argentine embassies called it "business as usual." "There is a job to be done," said one British official. "Obviously, you get on and do it. You don't start partying and leave the phones ringing."

In fact, last night at the annual British Embassy variety show called "Music Hall '82," there was deliberately no mention of the war. "It's not appropriate," said Sidney Langosch, director and producer of the show. Others connected with it felt the same way.

"We just hate that sort of thing," said Hazel Moore, a member of the management committee of the British Embassy Players, which puts on the show for staff and friends. "We don't want to see any young men killed. It's a terrible thing. There's no feeling of triumph at all. It never should have happened."

Earlier in the day, a spokesman at the Embassy said there was no celebration planned. Indeed, the huge embassy on Massachusetts Avenue was a model of restraint. "Are we going to jump up and down and run Union Jacks up the flagpole all day?" asked one embassy staffer. "No, we're not."

"It's not like World War II," said another staffer in the reception area. "It's just a small skirmish, really. Wars are nothing to celebrate."

Especially not a war that has been widely criticized for being foolish and costly. Certainly, no one seemed ready to break out champagne. The official embassy reaction, from spokesman Charles Anson, was simply described as "pleased and relieved."

But did the staff just sit there quietly when Prime Minister Margaret Thatcher was cheered in Parliament?

"No, of course not quietly," said Anson.

A couple of miles away, the mood at the Embassy of Argentina was one of unnatural calm. "We had lots of calls when the war broke out and when Secretary Haig gave his position," said Loures Flores, the receptionist who buzzes visitors in past the iron-grill door and answers phones. "But we haven't had many today. I guess it's not a novelty anymore."

Outside, one uniformed officer sat on his motorcycle studying Spanish from a tape recorder. "I expected more activity, myself," he said.

The Argentine press officer, Eduardo Jantus, downplayed the occasion as if--caught in a strange diplomatic time warp where no defeats and no losses are acknowledged--nothing momentous had happened.

"The war is not over," he said sitting back in his chair, a silk handkerchief tucked in his vest pocket. On the corner of his desk was a newspaper proclaiming an Argentine surrender. "We surrendered one city," Jantus said. "It's the surrender of one garrison. The battle is not the war. It's an episode."

Jantus took his usual morning jog before work and was preparing to go off to meetings with people he would not name.

"You want to see us in mourning?" he asked with a smile. He had just received a phone call from an Argentine urging the country to continue the fight, Jantus said.

"We lost a battle in 1833," he said. "We lost a battle yesterday." He sat back in his chair and shrugged. "The war continues. It's a question of integrity for our country. We never accepted this conquest by a foreign power."

He seemed to feel it might not last long. "The British conquered all of France," Jantus said, "and lost the war afterwards."

The flood of phone calls had been on Monday, Jantus said, the day that news of a surrender first surfaced. But even then, the routine of the embassy was not broken.

At 4 p.m. they had afternoon tea, a custom they denied had anything to do with the British. "Nobody can claim to have started afternoon tea," said Flores who took hers while reading a copy of "How to Make Love to a Man."

The calls that did come in yesterday morning were often sympathetic, said Flores. "I just had a call from an American offering support," she said yesterday. "Isn't that sweet? We have a whole notebook of American citizens who call. Some call long-distance. We have them from as far away as Arkansas or California." But aren't Americans sympathizing with the British? "Irish-Americans call," said Flores. "They say . . . 'We hate the British.' "

"Que hay?" What's happening? asked one woman soberly as Flores buzzed her into the embassy.

"Nada mas," Nothing more she answered.

Argentine Ambassador Esteban Takacs spent the morning in private meetings in his office and then outside the embassy.

Earlier Jantus returned from a meeting and rushed back out, on his way to a television interview, according to the receptionist. "There will be no statement," he said, getting into his car. "You have the government's word and you have the wires." He drove off, past a one-way sign at the corner of Q and New Hampshire that bears this prominent scrawl of graffito on the back side: "Cry for yourself, Argentina."

Back at the British Embassy, Ambassador Sir Nicholas Henderson was unavailable for comment, although he had done, as spokesman Anson put it, some of yesterday's "breakfast shows." Anson himself had been up since 5 a.m., answering a phone that was "ringing solid." Many calls were congratulatory. "As soon as I came in this morning, I took a call from California," he said, declining to specify whom it was from. "They were pleased."

On Monday, the day the cease-fire was first announced, the mood at the embassy was much the same. Another spokesman said he was also busy answering the phone.

But what about when the work was done? What then?

"I might have a gin and tonic," he said.