London — An apparent stalemate in the search for peace in the South Atlantic causes worried British politicians and citizens to ask how long President Reagan can remain ''evenhanded'' between Argentina and Britain.
Backbench Conservative members of Parliament and senior opposition Labour Party figures want the United States to take Britain's side, impose economic sanctions against Argentina, and persuade Japan to do the same.
Actual military aid is not expected, although pressure would grow for some US logistical backup services if a shooting war breaks out.
Tories such as prominent backbenchers Geoffrey Rippon say the ''natural'' US role is to keep the Atlantic alliance firm against the Soviet Union and to stand closer to Britain.
Denis Healey, the Labour Party's spokesman on foreign affairs (and former foreign secretary), said April 18 that by siding openly with Britain the US could allow UN Secretary-General Javier Perez de Cuellar to take over the role of mediator.
Hopefully he said this would pave the way for a UN administrator to run the Falkland Islands after the Argentine forces leave, in the same way the UN administered West New Guinea during talks between the Netherlands and Indonesia.
A UN administrator could also arrange for the Falkland Islanders themselves to vote on their future. The Thatcher government has said their wishes must be ''paramount.''
''The American people don't believe it is right for their president to be neutral between the aggressor and the victim of aggression,'' Mr. Healey said in a radio interview.
The Sunday Times April 18 cited British government sources as saying that Prime Minister Margaret Thatcher herself will expect President Reagan to show an ''iron fist'' if Argentina refuses to withdraw.
The pro-US Economist magazine considers it ''all but certain'' that if negotiations fail, the US will back Britain with economic sanctions against Argentina and perhaps logistical support for the British fleet.
Letting the Argentine junta know where the US would stand would be useful pressure, the magazine says:
''America's interest lies overwhelmingly in preventing British failure should mediation peter out. . . . Have-it-both-ways irresolution on the part of the US will lose British popular support for America's nuclear policies and deployment, and for its European, its NATO, and its Soviet policies. . . . The Russians have been fishing for geopolitical advantage . . . only because they sense American irresolution. . . .''
Comments like these increase pressure on Mrs. Thatcher, who continues to insist that Argentina must withdraw from the islands, and the British flag fly again, before negotiations begin.
In public she has had no choice but to praise US Secretary of State Alexander M. Haig Jr. for his mediation efforts.
Privately, government sources say it is impossible to ask the US for trade sanctions while Mr. Haig is trying to bring about a settlement, because it would undermine his position.
But talk here about the United States taking sides grows in volume as a diplomatic solution takes longer and longer to achieve. If Britain should find itself in a shooting war, demands for US support will grow heated and insistent as they were when Britain and France launched an attack on the Suez Canal Zone in 1956. At that time the US stayed out.
One fear is that if the US tries to stay out again, a sizable segment of British public opinion would not soon forgive it. The cornerstone of the Atlantic alliance could be weakened, and the only beneficiary would be the Soviet Union. Meanwhile, as the dispute lengthens, Britain is beginning to ask other questions.
One is whether Mrs. Thatcher may have to raise taxes to pay for a long blockade of the Falklands, including the logical next step -- an air blockade to prevent Argentine C-130 Hercules transport aircraft from resupplying invasion forces.
The government has already said it is looking at higher taxes as one way to raise any necessary extra revenue. Many of the task force costs would have been incurred anyway and will be met from the defense budget (about (STR)13 billion ( 83).
According to chief Treasury Secretary Leon Brittain April 9, that extra cost would have to be met in a ''noninflationary way'' -- that is, without extra government borrowing. The government has already requisitioned 26 civilian ships. The longer the task force is used, the more likely it is that indirect taxes could be raised here, or standard income taxes boosted temporarily.
The dispute occurred just as British inflation was falling toward single figures, world oil prices were down, and money supply figures were encouraging. The dispute has already sent share prices falling and has hardened some interest rates, raising the prospect that even a mild business recovery could see a rush to the banks for funds -- and interest rates go even higher.