A troubled, rain-soaked President Carter, head down, hand in pocket is on the cover of this week's Economist magazine. The influential weekly's headline reads, "It takes a worried man."

In a long analysis of the American dilemma, the editor of the Financial Times, Britain's most respected daily, declares that "central to the malaise is, of course, the performance of the president."

On a popular commercial television news program, the commentator asks an American correspondent, "Isn't Carter handicapped because his advisers all come from rural Georgia?"

The president's stock has plunged sharply in Britain in recent weeks, in a country that unashamedly admires most things American and regards the United States as its closest, most intimate ally.

The falling dollar and stock market, the remarkably publicized coal strike, the stalled energy program and the struggle for the Panama Canal treaties add up to a cumulative burden that worries and disturbs Britons at all levels.

The concern is deepened by a single fact: Britons, like other West Europeans, look to the United States for leadership, for protection against both economic and military disaster.

As the conservative Daily Telegraph asked in an editorial the other day:

"Is Carter a leader? Has Mr. Carter got what it takes?"

Some of the mood here is what Germans call schadenfreude , a delight in a big man's fall on a banana skin. For Britons who have spent 30 years getting over the falling pound, the dollar's troubles are a source of mingled anxiety and joy.

So Rupert Murdoch's tabloid Sun professes to be "shocked" at the spectacle of West Germans giving GIs good parcels. Perhaps protesting too much, the paper says, "We take no pleasure in Uncle Sam's temporary discomfiture."

"What is beginning to worry some people on this side of the Atlantic, about Mr. Carter," the paper said, "is the impression he seems to convey of being one and the same time omni-competent yet indecisive."

The Economist, a sympathetic and knowledgeable interpreter of the American scene, argues that there is always a clamor for presidential action when worries crowd in on the United States.

President Carter, who jumped in on too many issues during his first month in office, has learned to be more cautious. That he now looks a worried man should be reassuring to those who earlier thought him niave."

The real test of his leadership, the paper said, will be his ability to get his energy plan through congress - even if it is unlikely to cut U.S. demand for foreign oil by very much.

The prevailing view among thoughtful people here was expressed today by M. H. (Fredy) Fisher, the Financial Times editor, in his dispatch from New York.

"The puzzle is President Carter," Fisher wrote. "Is he, as his supporters would argue, the world leader who has quite deliberately addressed himself to the most fundamental and difficult issues.

"Or is he, as his critics would maintain, merely a brilliant campaigner who in office has turned out to be a Don Quixote, tilting at every windmill in sight?"

Some of the jeers are inextricably bound up with local concerns. Commenting on U.N. Ambassador Andrew Young's remark about a British "runout" on Rhodesia, the Daily Mail yesterday said that Young got his job only "because he helped Jimmy Carter corner the colored vote." The tabloid urges, "Give Andy Pandy the cap and Bells" of a "court jester of the White House cocktail circuit."

The mail, however, is concerned with winning readers from the Express and votes for the Conservative Party by inflaming racial passions.

Even after discounting for schadenfreude and parochialism, the concern here is deep-seated and genuine. It is clearly more intense among, those outside government, bankers, businessmen, writers, shop clerks and others that a correspondent meets in his daily rounds.

At 10 Downing Street, in the Foreign Office, at the Treasury, there is a more sympathetic understanding of the intractable problems with which Carter is dealing. Ministers and officials know there are no easy answers to Rhodesia, to the deficit imposed by the international oil combine, the Middle East, the stagnant economy and even coal miners. Britons too have wrestle with all these, and found little glory in the process.

Prime Minister James Callaghan is perhaps the most fervently pro-American Prime Minister since Winston Churchill. As finance minister, Callaghan regularly called on Washington to prop the $2.80 pound. He once told a visiting U.S. journalist that the first thing he did each morning was to sing God Bless America.

Above all, Callaghan has impressed on most commentators here that Carter is no rural innocent but an exceptionally intelligent man who masters briefs with a rare grasp of detail. The difficulty, as the Telegraph observed, lies elsewhere.

What is beginning to worry some people on this side of the Atlantic, about Mr. Carter," the paper said, "is the impression he seems to convey of being at one and the same time omincompetent yet indecisive."

The Economist, a sympathetic and knowledgeable interpreter of the American scene, argus that there is always a clamor for presidential action when worries crowd in on the United States.

"President Carter, who jumped in on too many issues during his first months in office, has learned to be more cautious. That he now looks a worried man should be reassuring to those who earlier thought him naive.

The prevailing view among thoughtful people here was expressed today by M. H. (Fredy) Fisher, the Financial Times Editor, in his dispatch from New York.

"The Puzzle is President Carter," Fisher wrote. "Is he, as his supporters would argue, the world leader who has quite deliberately addressed himself to the most fundamental and difficult issues?

"Or is he, as his critics would maintain, merely a brilliant campaigner who in office has turned out to be a Don Quixote, tilting at every windmill in sight?"