The United States and other NATO countries have a great deal of work to do to win the battle for European public opinion on nuclear defense, British and US officials here agree.
The visit by Vice-President George Bush, which ended Feb. 10, did nothing to abate the vigorous opposition of the European peace movement to the deployment of cruise and Pershing II medium-range missiles, and it appeared to leave government positions virtually unchanged.
Conservative governments in London and Bonn were still the strongest missile supporters. Italy looked reasonably firm so far, while Belgium and the Netherlands were far less certain.
Both Prime Minister Margaret Thatcher in London and Chancellor Helmut Kohl in Bonn face elections soon. NATO leaders think Mrs. Thatcher will win, but are much more worried about Mr. Kohl's chances. A conservative loss would create new uncertainty about deployment, though it is not yet clear whether a loss would cancel it altogether.
As Mr. Bush flew home Feb. 10, however, NATO sources said they remained optimistic that NATO would in fact begin to deploy cruise and Pershing II missiles by the end of the year in Britain.
They insisted that unless Moscow sees missiles actually being installed, it will make no further concessions in the Geneva missile reduction talks.
They repeated a point Mr. Bush himself made in Rome and elsewhere: ''What goes in can also come out'' - that is, if an agreement is reached, missiles already deployed can be withdrawn.
Mr. Bush made an enthusiastic public effort to blunt the appeal of the peace groups, to promote the idea that the US was a flexible but determined peacemaker , and to listen to Europeans' ideas and concerns.
He tried to spread the word that President Reagan, who still appears a rigid hard-liner to many Europeans, was ''not wedded'' to his stand of trying to eliminate all medium-range missiles. Without actually saying so, he left the impression that Mr. Reagan would settle for a compromise agreement in Geneva - but would continue to try to reach a complete elimination of missiles.
Many Europeans want an unequivocal pledge now to a compromise agreement. The peace groups want no cruises or Pershings anywhere in Europe at all. The US feels it cannot give either pledge and still be a credible negotiator in Geneva.
Repeatedly the vice-president asked the Kremlin to put forward its own proposals to ban all missiles and promoted President Reagan's call to eliminate missiles ''from the face of the earth.''
President Reagan's call, made in an open letter read by Mr. Bush in West Berlin, did not seem to impress many Europeans. Noting that the Kremlin had swiftly dismissed it, newspaper editorials and commentators called it theatrical and unrealistic.
On his last stop, in London Feb. 9 and 10, Mr. Bush warned that the Soviet aim was to ''decouple'' Western Europe from the cover of the US nuclear umbrella. He insisted that ''an attack on you is an attack on us.''
''The trouble is that Bush said nothing really new,'' said one NATO source who closely followed reactions to the visit.
''The alliance needs new rhetoric, a more convincing way to convince Europeans that we are flexible in our search for peace.''
A source close to Mrs. Thatcher said, ''We have a lot of work yet to do, and we started late in Britain because of the Falklands war last year. We are encouraged by Mr. Bush's visit. You will hear no criticism of President Reagan from us.''
According to a Gallup poll published in London Feb. 9, just over 50 percent of those asked did not want US-controlled bases on British soil. Two out of three, however, rejected unilateral disarmament, and a majority wanted both nuclear weapons and a strong nuclear defense.
A pro-NATO political analyst in Hamburg commented in a telephone interview, ''Bush made little impact here, and especially not with young people. He said nothing new and his open letter from Reagan in West Berlin (offering to meet Soviet leader Yuri Andropov to ban all missiles 'from the face of the earth') was just theatrical rhetoric.''
Neither the Netherlands nor Belgium has taken a final decision to deploy. Holland has not even begun physical preparations. Belgium, without fanfare, has.