A small group of antinuclear activists on both sides of the Atlantic will be watching closely when the House of Representatives votes on funding for the Reagan administration's defense bill.

Largely obscured among the MX missiles, nerve gas and Star Wars research is an item of special concern to them -- the expenditure of $1.1 billion for production of up to 925 new nuclear artillery shells to replace aging warheads in NATO's battlefield nuclear arsenal in Western Europe.

There appears to be little doubt that production of the shell, known as W82, will go ahead. But three activists from Britain's Campaign for Nuclear Disarmament and the Washington-based Federation of American Scientists take some comfort in the attention they have drawn to the issue. It embarrassed the two governments and inserted a new wedge of irritation and mistrust into defense aspects of the "special relationship" between them.

The W82 controversy is part of the larger issue of the deployment of thousands of battlefield nuclear weapons by North Atlantic Treaty Organization forces throughout continental Europe. Many have argued that these tactical weapons, including nuclear land mines and nuclear-equipped howitzers with ranges as short as 20 miles, are no longer useful under modern NATO deterrence strategy and its increasing emphasis on strengthening conventional forces.

Others, including many in both the U.S. and British governments, say that expansion of Soviet battlefield nuclear capability argues in favor of modernization, if not expansion, of NATO counterparts.

The story of the W82 shows how shared U.S. and British security interests sometimes run afoul of domestic political considerations. Equally, however, it is a case study of the daily work of the antinuclear lobby after the protest banners have come down and the demonstrators gone home.

The controversy revolves around whether, and when, the British government agreed that the new shells, long sought by the Pentagon, were needed and would deploy them in its own NATO artillery.

The Pentagon, seeking to overcome congressional resistance, repeatedly has said the British Defense Ministry wants them. Asked by Parliament to respond, the defense ministry here -- which deals gingerly with highly volatile public opinion on all nuclear issues -- has said the subject still is under consideration.

Over the past several months, both Parliament and Congress have, in effect, accused their respective defense establishments of lying.

Last March, for example, Assistant Defense Secretary Richard Wagner was asked by a congressional subcommittee about three separate statements by British defense officials indicating that European agreement on the W82 "is far less tangible and more tenuous . . . than we have been told."

"I would disagree," Wagner replied, noting that the British "probably" were loath to be explicit on the matter "for home consumption."

As a result of these and other Pentagon statements, British Defense Secretary Michael Heseltine told Parliament in writing last week that he is "not responsible for decisions taken by the United States government, or evidence given by their officials to the United States Congress." That evidence, Heseltine wrote, "appears to have taken a different view to mine."

The British, one knowledgeable American here said, are "extremely unhappy" over the subject. Inquiries about W82 among antinuclear groups usually bring referrals to Dan Charles of the Federation of American Scientists in Washington, who has followed the weapon's tortured path to production.

The useful thing about the W82 debate, Charles said in a telephone interview, is that it points out "several disconnects within the whole alliance process." It is these, he said, "that Marjorie, Dan and I have highlighted."

Dan Plesch and Marjorie Thompson are part of the Campaign for Nuclear Disarmament parliamentary lobby effort. Thompson, an American who worked as a U.S. congressional staff member before moving to London, described their role as "pushing the issue of democratization of military policy."

In essence, that means keeping abreast of nuclear weapons developments on both sides, studying technical journals and reports, and following what each government says on the public record in its home territory presumably outside the immediate hearing of the other.

Often, those statements do not agree. The principal work of Charles, Thompson and Plesch has been to organize a transatlantic exchange of information, supplying Congress with British material with which to confront the Pentagon and vice versa.

Once a confrontation takes place, the lobbying organizations also try to notify the media. Last week, copies of relevant testimony from both legislatures were supplied to British newspapers and the bureaus in London of major American newspapers.

"There are those of us whose job it is to know these things," said Charles of the W82 debate. "We instigate the question in Parliament, try to get the governments on record and point out where they contradict each other."

In Britain, the lobbyists deal directly with members of Parliament, claiming 150 parliamentary "sympathizers" -- most of them opposition Labor Party members.

The defense ministry answers, most of which are printed in the daily public record of Parliament, are supplied to Charles, who has a working relationship with a number of staff members in the offices of congressmen who have questioned battlefield nuclear weapons, including Edward J. Markey (D-Mass.) and Vincent Fazio (D-Calif.).

Both lobby groups agree that information is much more easily obtained in Washington, where law and tradition make government officials more forthcoming. Here, the law supports a tradition of higher government secrecy, which Washington-originated information can be used to question.

British officials emphasize that the current disagreement with Washington over replacing the old shells with the W82 does not necessarily mean the British are against it. "It is a consultation issue rather than the actual replacement issue itself," said one.

In the British view, there is a NATO agreement that a final decision on the shells would not be made until after a period of consultation that is not yet over, despite the demands of the U.S. budgetary process.

"In both political alliance and in public terms," the British official added, "it behooves the United States to be extra prudent on issues like consultation." The current situation "undermines the whole process of nuclear planning . . . and undermines public opinion" on such issues as ongoing deployment of intermediate-range weapons.

Most of the discussion of W82 dates from October 1983, when NATO defense ministers meeting in Canada decided to reduce the number of nuclear warheads in Europe but to make sure the remaining stockpile was both "survivable . . . and effective," code words for modernization. The ministers, according to their communique, "identified a range of possible improvements."

In nuclear-sensitive Europe, the reduction decision was greeted with approval. There was little discussion of the "possible improvements."

According to Heseltine, the 1983 agreement was a general one. "My colleagues and I," he wrote Parliament last week, "did not endorse any specific proposals for modernization of the stockpile -- nor were we invited to."

Instead, he said, they asked the supreme allied commander in Europe to come up with a study that was presented to NATO ministers meeting at the Nuclear Planning Group last March in Luxemburg. That study, Heseltine wrote, "is now being considered."