London — A period of rising concern lies ahead in the Falkland Islands dispute. US mediation efforts have reached an impasse, although hope for a peaceful solution has not yet been lost.
For the moment, US diplomacy has failed to bridge the gap between Britain and Argentina on the central, crucial issue of whose flag would fly over the islands between the time Argentina's troops withdrew and agreement was reached on a long-term settlement.
But just when the peace mission of Secretary of State Alexander M. Haig Jr. seemed to have been brought to a halt, a last-minute flurry of meetings and telephone calls in London April 13 enabled him to hold out at least a glimmer of optimism.
Choosing each word carefully, he said at London's Heathrow Airport that ''we have now received some new ideas.'' Leaving both sides to consider them, he flew back to Washington to report to President Reagan, and planned to pick up his mission again ''shortly'' by going on to Buenos Aires.
Whether he does so will depend on the next moves in both Britain and Argentina.
On the face of it, the atmosphere looks grim. Mr. Haig conceded that ''the whole situation in the region is dangerous, and increasingly so'' and said that finding a solution was a matter of ''great urgency.''
Each day that goes by without a settlement increases the risk that British submarines might encounter Argentine ships in the 200-mile blockade zone, in force since April 12. It also allows the Royal Navy task force to steam that much closer.
Britain is still relying on the size and power of its Navy to force Argentina off the islands, which they seized April 2. London has requisitioned six more civilian ships to reinforce the impression that it will open fire if it must. As the dispute drags on, the chance of British lives being lost increases, the cost to the British taxpayer rises, and the pressure on Mrs. Thatcher grows. She must succeed--or she will probably lose her job.
Mr. Haig had clearly been set back on April 13.
He had planned to fly to Buenos Aires in the morning, but what the US Embassy called ''complications'' kept him in London and eventually diverted him to Washington instead.
The ''complications'' arose during telephone calls Mr. Haig made to Argentine Foreign Minister Nicanor Costa Mendez on the night of April 12-13. Later, British diplomatic sources were emphatic in saying they amounted to Argentina reneging on earlier positions conveyed to Mr. Haig in Buenos Aires.
The British insisted that all the blame lay in Argentina. This raised the prospect of a split in the ruling junta, perhaps between Navy and Army, or of a division between the junta on one hand, and Mr. Costa Mendez on the other.
Meanwhile in Buenos Aires itself, the blame was being laid on British intransigence, with Foreign Secretary Francis Pym singled out for particular criticism.
Mystery grew about Mr. Haig's travel plans April 13, a day of confusion.
He held a somber 90-minute talk with Mrs. Thatcher in the morning. Later Mr. Pym said Mr. Haig had explained what had gone wrong in the night. In the afternoon Mr. Haig telephoned Buenos Aires, and Mr. Pym appeared unannounced late in the afternoon at Haig's hotel suite. Mr. Pym left saying his talks had been ''useful''--diplomatic jargon indicating that some progress had been made.
As Mr. Haig climbed into his limousine to go to the airport, it was still not known whether he was heading for Buenos Aires (signaling that diplomacy was still alive) or Washington (assumed to mean a grave setback).
At the airport Mr. Haig referred to difficulties that had changed his plans to go to Argentina. He did not indicate who had suggested the ''new ideas,'' and he put the best face he could on the impasse by saying his return to Washington would take place while both sides examined the ideas, and that he planned to go on to Buenos Aires ''shortly.''
It was widely believed here that Mr. Haig had brought from Argentina April 12 a plan under which Argentina would withdraw its troops, and then three flags would fly over the islands - British, Argentine, and United States. The British task force would turn around, and talks on the islands' future would swing into high gear.
But this idea seems to have failed. Reports from Buenos Aires suggest it has been rejected by the junta. Despite Mr. Pym's assertion that Argentina is to blame, there is no indication here that Mrs. Thatcher was prepared to settle for anything less than removal of the Argentine flag and troops.
Britain has apparently refused to accept Peru's suggestion of a 72-hour truce , or a reported Argentine offer to replace its occupation forces on the islands with civilians.
The British government shows determination to play its naval card as hard as it can. In well-publicized moves, it has requisitioned a 9,000-ton supply ship based in Aberdeen to support the task force. It has ordered four freezer-trawlers based in the British port of Hull to prepare to sail south, apparently as minesweepers. And it is outfitting the cruise liner Uganda in Naples as a hospital ship.