The extent of Britain's control over the use of U.S. cruise missiles to be based here has emerged as a major issue in the country's nuclear debate, with Prime Minister Margaret Thatcher under pressure to insist on a veto right for Britain should the United States ever decide to use the missiles.

Politicians from all parties, including Thatcher's Conservatives, have expressed concern--as one Tory put it--that missiles "based on British soil" will not be under "sovereign British control." As it stands, the 160 cruise missiles to be sited at two British bases will be U.S.-owned and -operated.

With deployment of cruise scheduled to begin later this year, widespread public unease over the missiles has revived the control question, first discussed three years ago. The United States then offered Britain a "dual-key" partnership, but it was turned down.

The problem is certain to be discussed with Vice President George Bush when he visits London Wednesday at the end of his European tour. While Thatcher remains a strong supporter of U.S. negotiating strategy on medium-range missiles in Europe, a number of recent British polls show a majority of Britons opposed the stationing of cruise missiles here.

Although the public remains in favor of maintaining Britain's own nuclear deterrent, the polls reflect considerable opposition--all across the political spectrum--to forgoing control over weapons fired from British territory. "The British public does not trust President Reagan's finger on the nuclear trigger," the London Sunday Times concluded in reporting a survey last month.

This poll, conducted by Market Opinion Research International, found that 93 percent of those questioned favored a dual key on cruise missiles, giving Britain direct physical responsibility for any launching.

"I fully understand the depth of feeling that exists on this issue," Michael Heseltine, the recently appointed defense minister, said under stiff questioning in Parliament last week. But he reaffirmed the government's view that existing arrangements with the United States, dating back 30 years, give Britain the restraints it needs over the firing of nuclear weapons.

As disclosed now in detail, the terms assert that while the United States has "certain bases" in Britain, their "use . . . in an emergency would be a matter for joint decision in light of circumstances at the time."

This is interpreted by the government, official sources said, as a sufficient safeguard to veto the firing of a missile. "Joint decision" on use of the bases, the sources said, clearly would also apply to the weapons located there. The government, Heseltine told Parliament, is satisfied that these measures are "fully effective."

Moreover, the arrangements are the same as those that apply to U.S. use of bases for nuclear-armed Poseidon submarines and F111 fighter-bombers. To the supporters of the dual-key arrangement, the difference appears to be that submarines and bombers carrying nuclear missiles would not actually launch them from British territory.

The concept of dual-key control dates back to the 1950s when Thor and Jupiter missiles were shared with the United States. British troops manned the missiles, with U.S. responsibility for the warheads.

During the 1979 NATO negotiations on the stationing of cruise and Pershing II missiles in Europe, the United States offered its allies joint ownership but was turned down for political and financial reasons. The British cost of purchasing the missiles could have been nearly $1 billion, according to The Economist. This was considered an unneccesary expense given the longstanding "joint decision" agreements.

The West Germans indicated that it would be politically divisive at home and needlessly provocative to the Soviets for Bonn to have, as the London Times put it, "even half a finger on the trigger." The Italians also considered a dual-key purchase of the weapons and decided against it.

The issue was raised by critics of the government's nuclear policies on the eve of Secretary of State George P. Shultz's visit to London last December. Shultz and Foreign Secretary Francis Pym said after meeting that there was "no particular likelihood" of any change in the existing arrangements. But the issue shows no sign of subsiding and influential voices are calling for some means of making British responsibility for the weapons more explicit.