Secretary of State Cyrus R. Vance was told today that Britain's move toward recognition of Zimbabwe-Rhodesia will be accomplished slowly to avoid increasing domestic American pressure on President Carter to follow the same path.

Vance was understood to have received this assurance during five hours of talks with Lord Carrington, foreign secretary in Prime Minister Margaret Thatcher's new Conservative government.

Carrington is understood to have said that steps to recognize the black majority government of Prime Minster-elect Abel Muzorewa will be slow and cautions and almost certainly will not produce a formal decision until after the meeting of British Commonwealth nations this August in Lusaka, Zambia.

Such a go-slow approach could be of considerable help to Carter, who is under heavy congressional pressure to reverse U.S. policy by lifting economic sanctions against Rhodesia and endorsing the legitimacy of the Muzorewa government, which is to take office before June 1.

Last week, the Senate overwhelmingly passed ending a resolution calling for the immediate ending of sanctions. The Carter adminstration's response has been to plead for more time to follow the mandate of 1978 legislation requiring Carter to determine whether Rhodesia's recent elections, boycotted by antigovernment nationalists were reasonably free and fair.

The administration's pleas would be undermined seriously if Britain, whose previous Labor Party government had been allied with Washington in a joint plan for dealing with Rhodesia, broke ranks precipitiously. Fears of a British withdrawl from the Anglo-American plan were intensified by the election of Thatcher, who promised to recognize the Muzorewa government if British observers determined it was elected fairly.

Thatcher's observers formally reported last week that, despite some reservations, they found the Rhodesian election to be sufficiently free and fair to justify recognition. Although the observers' lengthy report has not yet been made public, U.S. officials have a copy.

That brought Vance here at the start of a two-week European and Middle Eastern trip to seek ways of preventing an end of U.S.-British cooperation on Rhodesia.

Sources on both sides said Rhodesia took about 90 minutes of today's meeting and that the two ministers will discuss further, probably on Wednesday. Except for describing the meeting as "fruitful" and "profitable," neither Vance nor Carrington would talk publicly about the sustance of their discussions.

But it is known that Carrington, while emphasizing his government's commitment to a change in policy toward Rhodesia, expressed understanding for Carter's problems and a desire to plot Britain's course in a way that will not aggravate them.

U.S. sources said, though, that it would be misleading to expect the meetings here to produce agreement on a new joint strategy. For one thing, the sources noted, the direction of U.S. policy toward Rhodesia will not be known until Carter makes the determination required by Congress about whether the Rhodesian election merits lifting of sanctions.

Confronting Carter is the question of whether he will have to abandon his policy of seeking to include all the factions fighting a civil war in Rhodesia in a solution to the strife that has troubled the southern Africa country since it broke away from Britain in 1965.

Because of a belief that no lasting solution can be found without the participation of guerrialla forces fighting from bases in neighboring black African countries, the United States and the former Labor government here refused to recognize the elections leading to a biracial government with a black majority.

Instead, the Anglo-American plan called for a peace conference of all factions, including the guerillas, to write a new constitution and hold one-man, one-vote elections under international supercision. Neither Washington nor London was able to gain acceptance of the plan.

The Thatcher government's strategy is evolving in what one knowledgeable source characterized today as "a series of tentative moves toward the ultimate objective" of recognizing the Muzorewa government and lifting British economic sanctions. "They are testing the water with their toes," the source said.

The next step is the setting up of "permanent arrangements for continuing government, which the British Foreign Office announced last week. Lord Carrington is expected to provide more details in a speech to the House of Lords on Tuesday.

Informed sources said British will not be basing an ambassador-like envoy in Salisbury, but rather that one or more senior, London-based Foreign Office diplomats will visits Salisbury for periods of up to several weeks at a time.

Both the United Nations and the Commonwealth have refused to recognize the recent election result because the internal settlement constitution reserved a disproportionate amount of power for Rhodesia's whites, who will continue to control the Army and much of the civil service.

The high commissioners of the other 35 Commonwealth nations - Ccanada, Australia, New Zealand and nonwhite former British colonies in the Caribbean, Africa and Asia - called an emergency meeting here earlier this week to warn Britain not to recognize the Muzorewa government. They called the Rhodesian election a "gigantic faraud."

Nigeria, Britain's most important trading partner in Africa, has threatened to leave the Commonwealth and take other "appropriate actions" if Britain restores Rhodesia to legality and ends economic sanctions against it.

U.S. sources said Vance and Carrington also touched on problems of the Western alliance, the second U.S.-Soviet strategic arms limitation treaty (SALT II), the Middle East, the Far East and the Carribbean.

One U.S. source described the talks as "very full and very amiable" and said they seemed to "reaffirm the mutuatlity of interests and approaches to most issues."

U.S. sources said Vance was particularly gratified by Carrington's assurances that his government fully supports SALT II. The treaty faces a tough Senate fight and ther was some concern in Washington that a Conservative government here might view it with suspicion.