Ambassador Anne Armstrong has now gone home and, against all the odds, she will be genuinely missed.

She has been the first U.S. ambassador here in years to make any impact on either ordinary Britons or high policy. The near unanimous verdict of the professionals is that she has been a strong and useful force.

One veteran and hard-bitten U.S. diplomat observed the other day:

"When she came here, I thought this effusive Texan ahd to be a fraud. But I have grown fond of her. I even admire her."

Armstrong, 49, applied what another professional called "intelligence, hard work and a warm, attractive personality" to a task that it often more ceremonial than substantive. She and her embassy are crideited by Britons with playing a substantial part in overcoming the U.S. Treasury's reluctance to erect a safety net, the $3 billion international credit, to prop the pound.

Ostensibly, she is going back to Texas to help her husband Tobin run their huge ranch. But as a former co-chairman of the Republican Party and Cabinet-level counselor. Armstrong will soon be back in the thick of national politics.

The only serious question is whether she will make a stab at the governor's mansion or the U.S. Senate. Asked the other day how she would advise and aspiring politician. Armstrong grinned and said, "I'd run the race I could win."

Then she began soeculating out loud about the strength of Democratic Gov. Dolph Briscoe, suggesting that anyone who could cust him would have a major claim on national attention.

Armstron came here just a year ago with one big advantage. She was following two very different Republican ambassadors who had made virtually no dent on British life, Walter Annenberg and Elliot Richardson.

Armstrong was something else. A month after she came, she was the first U.S. ambassador to visit and its three-sided war among British soldiers and Protestant and Catholic murder gangfs. She spent two days there, jumping out of her bullet-proof Cadillac to talk with citizens on the embattled ground. She more or less achieved her aim, convincing Ulstyer's suspcicious sectarians that the U.S. government sternly opposed the Irish-Americans who are financing the Irish Republican Army (IRA).

She had to work hard at overcoming a stereotyped view of right-wing Texans, at winning a measure of credibility from Britain's socialist leaders. Most here agree she did it, by relent-less questioning of her aides, thorugh hours of preparatory reading and all manner aof careful, perssonal follow-up.

Winning Labor ministers in search of U.S. aid may not have been all that hard. But close observes credit her as well with charming Len Murray, head of the Trades Union Congress (TUC), the British AFL-CIO, and Jack Jones, the powerful leader of the transpot workers, Characteristically, Jones, Likeliest successor, Moss Evans, wass a guest at Armstrong's last official dinner.

An ambassador at a major embassy in an age of instant transaltantic communication rately has much effect on policy. In Henry Kissinger's State Department, career diplomats were routinely bypassed.

But Kissinger could not ignore Armstrong, one diplomat explained, partly because "she, like Richardson, was a power in the Republican Party and a potential Vice President and partly becaus he simply like her."

Last summer, Kissinger and the British ambassador in Washington, Sir Peter Ramsbotham, were working almost alone on the U.S. British strategy for southern Africa. An annoyed Armstrong insisted on putting herself - and the London governnment - into the picture and called Kissinger directly.

"He rolled over," one diplomat recalled in satisfaction.

More important were the complex negotiations over the $3.9 billion loan from the International Monetary Fund for a Britain then suffering from a falling pound. U.S. Treasury Secretary William E. Simon, backed by Chancellor Helmut Schmidt of West Germany demanded a harsh squeeze on British spending as the price for the loan.

Prime Minister James Callaghan's Cabinet answered that this was not only self-destructive economics but would also imperil hopes of winning labor's assent here to further wage restraint. Simon and Scdmidt would have won the argument by default, escept that Kissinger, uneasy with economics, decided to make a pitch on political grounds.

The embassy here credits Armstrong with persuading Kissinger to see the loan in a poltical context and to exert counterpressure against Simon and Schmidt. In the end, a conpromise was stuck. Britain got the money and had to undergo only a modest deflation. Moreover, as a sweetener, Kissinger bested, Simon and won agreement on the additional $3 billion safety net.

Now that the pound is rising on the prospect of a big flow of North Sea oil, the deflationary damage imposed by Simon, Schmidt and the IMF will soon be undone. The government here is expected to adopt a mildly expansionary budget at the end March.

Armstrong returns to Texas with a fresh perspective of her own. She came here more or less sympathetic to the Reagan wing to the Republican party and goes back as a thoroughly pragmatic politician. She has learned, as she will tell any audience, that the durability of free institutions is not fully reflected in the gross national product, that economics cannot measure all the strengths of a society.

She has learned, too, that lebels like "socialist" mean little in Western Europe, that problems are solved by realism rather than doctrine.

John Grigg in the weekly Spectator has already nominated Armstrong as the first woman President of the United States.