On the global circuit of elite foreign and defense policy think tanks, where power is endlessly analyzed for establishment use, Britain's leading entry is the Royal Institute of International Affairs, better known as Chatham House, the name of its elegant, 18th century premises on London's St. James's Square.

Its roster of distinguished mentors and patrons, beginning with the queen and including the heads of government of Britain and every one of the Commonwealth countries, outstrips even New York's Council on Foreign Relations in the blue-ribbon league.

Yet today, like so many other such august institutions in British life, Chatham House is undergoing something of a crisis of confidence and identity, struggling to maintain its standards and prestige in what are straitened circumstances. With less money than it needs and less influence than it (or its host country) once had, the Royal Institute is lacking the crucial tender that provides a measure of think tanks wherever they are.

"Chatham House is being squeezed," said one of its most ardent supporters, "and as its position falters, people are less keen on giving funds, which makes it less able to do good work."

Beneath the solemn portraits and down musty corridors, there is no visible hand wringing over the problems, no special passing of the hat. Indeed, the main difficulty, according to critics, is a deficiency of the sort of invigorating, anxious tension that gives less venerable outfits a boost. Meetings are duly held--although most lunches have been pared down to near-starvation rations of cucumber sandwiches--research papers and journals are churned out. But their impact is rarely what it should be. DAVID WATT, the director in recent years, is, he willingly admits, a shy person, for whom it is a struggle to be unceasingly glad handing benefactors and the prickly community that Chatham House serves. Watt said important studies have been completed lately on economic, energy and East-West topics and is proud of such innovations as lucrative conferences where business participants pay for knowledge.

Still, with some relief, Watt said he will return to full-time writing later this year.

After a search in which dozens of names were considered, the challenge of lending Chatham House more "pizazz"--to use a plainly distasteful American term that has, nonetheless, been locally adopted--is falling to Adm. Sir James Eberle, who was, until March, NATO's top naval officer. At 56, Eberle is so committed to vigor that he has opted for taking an apartment just outside the All-England Tennis club at Wimbledon, of which he is one of only 375 select members, rather than living closer to the job. THE ADMIRAL was chosen last week from a list of eight finalists that included Peter Jay, Britain's ambassador to Washington during the Carter administration, Roderick MacFarquahr, a Sinologist who will now take a tenured chair at Harvard and several prominent politicians. Instead of a high flier type with a media background like Jay or a scholar like MacFarquahr, the search committee clearly went for experience in making things shipshape.

Eberle is the sort of unusual figure among the British ruling classes, said Lawrence Freedman, a professor of war studies at Kings College, who can look a Cabinet minister or a business magnate in the eye and ask for money. Coming from a military man, the crassness of such hustling seems, somehow, less demeaning. CHATHAM HOUSE PREFERS not to think of itself as having competitors. But less than a mile away, on the periphery of Covent Garden's boutiques, restaurants and theaters, resides the International Institute for Strategic Studies. Although its subject matter is more narrowly focused than the Royal Institute's, comparisons are inevitable.

IISS, which is only 25 years old and maintains a small staff at a headquarters that is utilitarian at best, seems to be flourishing. Its annual Strategic Survey and report on the worldwide military balance are regarded as the last word. The yearly conferences it holds in different countries have become to strategic specialists what conventions are to Shriners, an indispensable and jovial confab.

Any think tank that is not subsidized by a government has to scramble to stay solvent. At the moment though, IISS is showing a modest budget surplus and is embarked on a fund-raising drive to match a $1.5 million grant from the Ford Foundation.

IISS's main difference with Chatham House is that it is international. Its present director, Robert O'Neill, is an Australian. Its last director was West German. The deputy director by tradition is British and the assistant director is American.

As a vestige of Britain's imperial period, Chatham House has faced--and not yet really overcome--the adjustment to this country's reduced role on the international stage. By contrast, the success of the IISS, its friends say, demonstrates the potential benfits of a British setting, spiritually midway between the United States and Europe.