Britain took a deep breath last week and asked itself: Can there really be life after Margaret Thatcher?

On Tuesday, Britain's prime minister faces the first serious challenge to her leadership since she took office in 1979. Conservative members of Parliament will vote on whether to re-elect Mrs. Thatcher as party leader or ditch her in favor of Michael Heseltine, the dashing former defense secretary. Unless she wins decisively, her premiership will be gravely weakened. She may not even survive, and that is something which should be of interest to all Americans, especially George Bush.

Under Thatcher's leadership, Britain (with the possible exception of Turkey) has proved to be Bush's most steadfast European ally in the Gulf crisis. While a fickle Congress has wavered and other members of the multinational coalition against Iraq have wobbled, Thatcher has stood so firm against compromise with Saddam Hussein that some British officials are worried about an "anti-Brit" backlash in the event of war.

But the implications of Thatcher's possible departure from office go far beyond the immediate crisis. Some predict it would be similar to the shock experienced after Winston Churchill went down to defeat in the 1945 elections. Whatever Tuesday's outcome, it is safe to say that U.S. policy-makers have barely begun to plan for the post-Thatcher era. Among the matters with which they must deal:

Thatcher remains one of NATO's most ardent supporters -- in contrast to the skeptical French and the equivocal Germans. If she is forced out, what will this mean for the Bush administration's plans to reform in order to preserve the Atlantic alliance?

The U.S. has long relied on Thatcher to counter the threat of an inward-looking "Fortress Europe" created by the European Community's move toward a single market by 1992. Without Thatcher, who will act as Washington's surrogate spokesman in the new Europe?

Thatcher's aura of invincibility goes back to 1975, when she acquired the nickname "The Iron Maiden" during her successful campaign to topple Edward Heath as leader of the Conservative Party.

She has turned her innate stubbornness and obstinacy over the years into virtues, standing defiant and ultimately triumphant against threats real and imagined: the British trade unions, the Looney Left, Gen. Galtieri of Argentina and, less successfully, Jacques Delors, president of the European Commission and the man leading the drive towards deeper European political and economic integration, at the (alleged) expense of British sovereignty.

The prime minister's grasp on power has appeared so unshakeable that she has been able to dispel almost all doubts about her staying power. Her ruthlessness in removing opponents would make Lucrezia Borgia blush. Not one member of her original 1979 cabinet remains in place today; the last survivor, Sir Geoffrey Howe, resigned this month on the issue of Britain's future in Europe, precipitating the current crisis.

Her present difficulties do not appear to have fazed her. She has passed word to the Bush administration that Sir Geoffrey's resignation and Heseltine's challenge are just more examples of "the boys acting up again." This may explain the breezy but short-sighted response by one State Department official last week: "We've been through these things before, it's nothing new."

Wrong. The last change in Tory leadership took place before Jimmy Carter was in the White House. Britain stands transformed by the intervening years. The Labour Party (on paper, at least) has turned into a party of militant moderates, the old generation of union barons having been obliterated. Billions of dollars of state assets, such as gas, telecommunications, steel and housing have been transferred into private hands.

Abroad, her influence has been equally profound. It was Mrs. Thatcher who first singled out Mikhail Gorbachev as a reformer, proclaiming that here was a man the West could do business with. From 1985 to 1988, Thatcher positioned herself astutely between Moscow and Washington, smoothing over mutual suspicion in the interests of progress on arms control, human rights and political reform in Eastern Europe.

But the "special relationship" cultivated between Thatcher and Ronald Reagan was more a friendship based on personal warmth and ideological cameraderie than a strategic power equation. When George Bush moved into the White House, the mood changed. The president and Secretary of State James Baker set a new rule: this administration had no intention of being pushed around by "Maggie" the way its predecessor had often been.

America's new -- and necessary -- preoccupation with Germany made matters worse. Bush signalled he was unhappy about Thatcher's public doubts over the speed of unification, and Baker was disturbed by her truculence towards European economic unity. "It was more a question of tone rather than substance," recalls one British official, "but we risked becoming marginalized."

Iraq's invasion of Kuwait on Aug. 2 proved the turning point, and as Fred Barnes first reported in The New Republic, Thatcher's influence was critical in the early formation of policy and crisis management.

Both the prime minister and Bush were due to give speeches at an Aspen Institute symposium on Aug. 2, but the real reason Bush went was to consult with her. In their talks, Thatcher set out the stakes involved, including the analogy with the 1930s when the rest of world chose to appease rather than confront Hitler. "She was a big influence on the basic decision he had to make: What are the U.S. and Western interests in the Gulf, are they long-term or short-term, and are they worth defending?" a Bush aide who attended the meeting told Barnes.

In recent weeks, Britain -- that is, Mrs. Thatcher -- has become increasingly restive about the assorted diplomatic maneuverings aimed at reaching a "political solution" with Saddam Hussein. These peaked in late October, when Bush went through possibly the worst week of his presidency as the budget agreement fell apart and he found himself shunned by Republican candidates on the stump. In London's view, Bush's eye was drifting off the ball. The credibility of the multinational military force -- vital to persuading Saddam to withdraw from Kuwait -- was eroding.

It would be an exaggeration to say that Thatcher was responsible for persuading Bush to double the size of U.S. forces in the Gulf, but it is certainly true that Thatcher believes in making the threat of force as realistic as possible.

Occasionally she sounds a little too like the Warrior Queen. Only last week in the House of Commons she virtually committed the coalition to war, short of an imminent withdrawal by Iraq from Kuwait. But for the most part, Bush has been glad to point to Britain as a stalwart ally willing to deploy Britain's Desert Rats to fight alongside Americans -- if it comes to war. The worry in Washington is not so much that a successor would ditch support for the policy, but more that the demonstrative and unequivocal tone would disappear, leaving Europe looking a good deal less solid in its support.

What are the chances of a change in leadership? The first point to stress is that Tuesday's ballot is not a U.S.-style presidential election for the post of prime minister; rather it is a contest for the leadership of the Conservative Party. Only because the Conservatives are the majority ruling party will the new leader become de facto the new prime minister.

If Thatcher is to win in the first round, she needs to obtain an overall majority among Conservative MPs and win 15 percent more votes than any other candidates; this would require a minimum of 187 votes, with a margin of 56 votes more than the nearest challenger. If Thatcher and Heseltine each fail to achieve these targets, and a second ballot becomes necessary, other candidates could come forward. At this point, the contest could become extremely interesting.

Heseltine, who has coveted the job of prime minister since he was an undergraduate at Oxford, has let it be known that he has the votes of 100 MPs in his pocket. This is nowhere near enough to win a clear-cut victory; but he and his supporters hope that enough MPs (and ministers) are sufficiently dissatisfied with Thatcher's domineering leadership, and sufficiently worried about their re-election prospects, that they will abstain. This would force a second ballot, where only a bare 51 percent majority is needed to claim victory.

The question only Thatcher can answer is whether she will bow out if the contest goes to a second round, or whether she will fight to the bitter end -- even at the risk of splitting the party. And if she does stand aside, will the Heseltine bandwagon prove unstoppable? Or will Foreign Secretary Douglas Hurd, or, less likely, Treasury Secretary John Major, throw their hats into the ring?

These politicians are little-known in the United States, to say the least. Major and Hurd are mere blips on the horizon. Heseltine looms a little larger, but only because of the attention paid by press and television to his blond mane, his handsome looks and his suspect temperament. Such coverage fails to prepare the American public for political alternatives.

Last week, American press and television gave sparse coverage to the event which may have signaled the beginning of the end: Sir Geoffrey Howe's resignation speech to the House of Commons.

The speech was short and devastating. Its skill lay in combining doubts about Thatcher's autocratic leadership style with the substantive issue of Britain's future in Europe, specifically the prime minister's reluctance to countenance Britain's entry into the European Monetary System's exchange rate mechanism, a refusal for which Sir Geoffrey blamed much of the country's present economic problems.

With this kind of skimpy treatment, it is little wonder that most Americans have little idea what all the fuss is about in London, still less that the lady once described as "The Great She-Elephant" and "She-Who-Must-Be-Obeyed" is fighting for her political survival, and no inkling at all of what her passing might mean for them.

Lionel Barber is the Washington correspondent for the Financial Times of London.