LONDON — BRITAIN'S ruling Conservatives have won a breathing space in their battle to turn back a determined challenge by the Labour opposition, but the position of Prime Minister Margaret Thatcher remains dangerously exposed to attack from within her own party. The results of the May 3 local elections produced two major shocks for the Labour Party, both of them in London. The Conservatives swept to unexpectedly heavy victories in two ``flagship'' areas - Westminster and Wandsworth - where their low taxation strategy appealed to voters.
This helped to cushion gains by Labour of more than 300 local government seats in other parts of the country. It did not however do much to boost Mrs. Thatcher's popularity or her standing with critical supporters in parliament.
After Thursday's vote, a public opinion poll showed that Labour's big lead over the Conservatives in recent weeks had shortened from 20 to 13 percent. But 64 percent said they believed that Thatcher should stand down before the next general election, due in two years.
This view was rammed home by Barry Porter, a former Thatcher parliamentary loyalist, who said: ``I would like to say to the prime minister: `Thank you and good-bye - enjoy yourself.'''
Peter Walker, who stepped down from the Thatcher Cabinet on May 4 after serving in all Conservative administrations since 1970, said Thatcher's attachment to an unpopular poll tax levied on adult citizens had damaged her party's political prospects.
In the immediate wake of the May 3 vote, all leading political parties were claiming that the results favored them. Neil Kinnock, the Labour leader, said that they indicated that his party could expect a comfortable parliamentary majority after a general election.
Kenneth Baker, the Conservative Party chairman, derided Kinnock's claim and insisted that the result showed Labour's appeal to voters was waning. Come a general election, he said, the Conservatives could expect to win.
Paddy Ashdown, leader of the small Social and Liberal Democratic party, claimed that the outcome had helped to consolidate the SLD's position in ``three-party politics.''
According to a post-poll analysis in the London Sunday Times, the breakdown of votes on May 3 was Labour 43 percent, Conservatives 32 percent, SLD 16 percent. This, the paper said, would return a House of Commons in which Labour would have an overall majority of 68 seats.
What was conspicuously absent, as each of the parties tried to put the best possible construction on the outcome, was a wave of vocal support from leading Conservatives for the prime minister's own position as leader.
Anthony Beaumont-Dark, a leading Conservative member of Parliament, said: ``The results were good only relatively because they were not the disaster predicted. But they were nothing to be satisfied about.''
An ex-minister who has been critical of Thatcher said: ``The results have applied a brake to the leadership issue, but there is still a momentum.''
Another ex-minister emphasized a broader point: ``By stressing the victories in Wandsworth and Westminster, the prime minister and the party chairman are neglecting to note the message sent them by voters in other parts of the country.''
He referred to an opinion poll which asked who, regardless of party preference, would make the best prime minister. Offered a choice between Thatcher and Neil Kinnock, 53 percent of those surveyed preferred Kinnock.
Given a choice between Kinnock and Michael Heseltine, the Conservative ex-minister considered most likely to challenge Thatcher, 53 percent of the same sample opted for Mr. Heseltine.
This suggests that if the Conservatives wish to ensure victory over Labour at a general election, they would do well to think hard about the electoral appeal of Thatcher, who at the weekend marked her 11th anniversary as prime minister.
She is unlikely to enjoy a respite from the pressures that have been piling up on her. The monthly inflation figures will be issued this Friday and there have been forecasts from City of London financial sources that they will top 10 percent for the first time in eight years.
The main reason for the expected sharp rise of 8.1 percent will be the poll tax.