The World Health Organization, always cautious, acted more forcefully and faster than many national governments. But President Trump has decided to cut off U.S. funding to the organization.

On Jan. 22, two days after Chinese officials first publicized the serious threat posed by the new virus ravaging the city of Wuhan, the chief of the World Health Organization held the first of what would be months of almost daily media briefings, sounding the alarm, telling the world to take the outbreak seriously.

But with its officials divided, the W.H.O., still seeing no evidence of sustained spread of the virus outside of China, declined the next day to declare a global public health emergency. A week later, the organization reversed course and made the declaration.

Those early days of the epidemic illustrated the strengths and weaknesses of the W.H.O., an arm of the United Nations that is now under fire by President Trump, who on Tuesday ordered a cutoff of American funding to the organization.

With limited, constantly shifting information to go on, the W.H.O. showed an early, consistent determination to treat the new contagion like the threat it would become, and to persuade others to do the same. At the same time, the organization repeatedly praised China, acting and speaking with a political caution born of being an arm of the United Nations, with few resources of its own, unable to do its work without international cooperation.

Mr. Trump, deflecting criticism that his own handling of the crisis left the United States unprepared, accused the W.H.O. of mismanaging it, called the organization “very China-centric” and said it had “pushed China’s misinformation.”

But a close look at the record shows that the W.H.O. acted with greater foresight and speed than many national governments, and more than it had shown in previous epidemics. And while it made mistakes, there is little evidence that the W.H.O. is responsible for the disasters that have unfolded in Europe and then the United States.

The W.H.O. needs the support of its international members to accomplish anything — it has no authority over any territory, it cannot go anywhere uninvited, and it relies on member countries for its funding. All it can offer is expertise and coordination — and even most of that is borrowed from charities and member nations.

The W.H.O. has drawn criticism as being too close to Beijing — a charge that grew louder as the agency repeatedly praised China for cooperation and transparency that others said were lacking. China’s harsh approach to containing the virus drew some early criticism from human rights activists, but it proved effective and has since been adopted by many other countries.

A crucial turning point in the pandemic came on Jan. 20, after China’s central government sent the country’s most famous epidemiologist, Zhong Nanshan, to Wuhan to investigate the new coronavirus racing through that city of 11 million people. Dr. Zhong delivered a startling message on national television: Local officials had covered up the seriousness of the outbreak, the contagion spread quickly between people, doctors were dying and everyone should avoid the city.

Dr. Zhong, an eccentric 83-year-old who led the fight against the SARS outbreak of 2002 and 2003, was one of few people in China with enough standing to effectively call Wuhan’s mayor, Zhou Xianwang, a rising official in the Communist Party, a liar.

Mr. Zhou, eager to see no disruption in his plans for a local party congress from Jan. 11 to 17 and a potluck dinner for 40,000 families on Jan. 18, appears to have had his police and local health officials close the seafood market, threaten doctors and assure the public that there was little or no transmission.

Less than three days after Dr. Zhong’s warning was broadcast, China locked down the city, preventing anyone from entering or leaving and imposing strict rules on movement within it — conditions it would later extend far behind Wuhan, encompassing tens of millions of people.

The national government reacted in force, punishing local officials, declaring that anyone who hid the epidemic would be “forever nailed to history’s pillar of shame,” and deploying tens of thousands of soldiers, medical workers and contact tracers.

It was the day of the lockdown that the W.H.O. at first declined to declare a global emergency, its officials split and expressing concern about identifying a particular country as a threat, and about the impact of such a declaration on people in China. Such caution is a standard — if often frustrating — fact of life for United Nations agencies, which operate by consensus and have usually avoided even a hint of criticizing nations directly.

Despite Dr. Zhong’s warning about human-to-human transmission, Tedros Adhanom Ghebreyesus, the W.H.O.’s director-general, said there was not yet any evidence of sustained transmission outside China.

“That doesn’t mean it won’t happen,” Dr. Tedros said.

“Make no mistake,” he added. “This is an emergency in China, but it has not yet become a global health emergency. It may yet become one.”

The W.H.O. was still trying to persuade China to allow a team of its experts to visit and investigate, which did not occur until more than three weeks later. And the threat to the rest of the world on Jan. 23 was not yet clear — only about 800 cases and 25 deaths had been reported, with only a handful of infections and no deaths reported outside China.

“In retrospect, we all wonder if something else could have been done to prevent the spread we saw internationally early on, and if W.H.O. could have been more aggressive sooner as an impartial judge of the China effort,” said Dr. Peter Rabinowitz, co-director of the MetaCenter for Pandemic Preparedness and Global Health Security at the University of Washington.

Amir Attaran, a public health and law professor at the University of Ottawa, said, “Clearly a decision was taken by Dr. Tedros and the organization to bite their tongues, and to coax China out of its shell, which was partially successful.”

“That in no way supports Trump’s accusation,” he added. “The president is scapegoating, dishonestly.”

Indeed, significant shortcomings in the administration’s response arose from a failure to follow W.H.O. advice.

The Centers for Disease Control and Prevention bungled the rollout of diagnostic tests in the United States, even as the W.H.O. was urging every nation to implement widespread testing. And the White House was slow to endorse stay-home restrictions and other forms of social distancing, even after the W.H.O. advised these measures were working in China.

It is impossible to know whether the nations of the world would have acted sooner if the W.H.O. had called the epidemic a global emergency, a declaration with great public relations weight, a week earlier than it did.

But day after day, Dr. Tedros, in his rambling style, was delivering less formal warnings, telling countries to contain the virus while it was still possible, to do testing and contact tracing, and isolate those who might be infected. “We have a window of opportunity to stop this virus,” he often said, “but that window is rapidly closing.”

In fact, the organization had already taken steps to address the coronavirus, even before Dr. Zhong’s awful revelation, drawing attention to the mysterious outbreak.

On Jan. 12, Chinese scientists published the genome of the virus, and the W.H.O. asked a team in Berlin to use that information to develop a diagnostic test. Just four days later, they produced a test and the W.H.O. posted online a blueprint that any laboratory around the world could use to duplicate it.

On Jan. 21, China shared materials for its test with the W.H.O., providing another template for others to use.

Some countries and research institutions followed the German blueprint, while others, like the C.D.C., insisted on producing their own tests. But a flaw in the initial C.D.C. test, and the agency’s slowness in approving testing by labs other than its own, contributed to weeks of delay in widespread testing in the United States.

In late January, Mr. Trump praised China’s efforts. Now, officials in his administration accuse China of concealing the extent of the epidemic, even after the crackdown on Wuhan, and the W.H.O. of being complicit in the deception. They say that lulled the West into taking the virus less seriously than it should have.

Larry Gostin, director of the W.H.O.’s Center on Global Health Law, said the organization relied too heavily on the initial assertions out of Wuhan that there was little or no human transmission of the virus.

“The charitable way to look at this is that W.H.O. simply had no means to verify what was happening on the ground,” he said. “The less charitable way to view it is that the W.H.O. didn’t do enough to independently verify what China was saying, and took China at face value.”

The W.H.O. was initially wary of China’s internal travel restrictions, but endorsed the strategy after it showed signs of working.

“Right now, the strategic and tactical approach in China is the correct one,” Dr. Michael Ryan, the W.H.O.’s chief of emergency response, said on Feb. 18. “You can argue whether these measures are excessive or restrictive on people, but there is an awful lot at stake here in terms of public health — not only the public health of China but of all people in the world.”

A W.H.O. team — including two Americans, from the C.D.C. and the National Institutes of Health — did visit China in mid-February for more than a week, and its leaders said they were given wide latitude to travel, visit facilities and talk with people.

Whether or not China’s central government intentionally misstated the scale of the crisis, incomplete reporting has been seen in every other hard-hit country. France, Italy and Britain have all acknowledged seriously undercounting cases and deaths among people who were never hospitalized, particularly people in nursing and retirement homes.

New York City this week reported 3,700 deaths it had not previously counted, in people who were never tested. The United States generally leaves it to local coroners whether to test bodies for the virus, and many lack the capacity to do so.

In the early going, China was operating in a fog, unsure of what it was dealing with, while its resources in and around Wuhan were overwhelmed. People died or recovered at home without ever being treated or tested. Official figures excluded, then included, then excluded again people who had symptoms but had never been tested.

On Jan. 31 — a day after the W.H.O.’s emergency declaration — President Trump moved to restrict travel from China, and he has since boasted that he took action before other heads of state, which was crucial in protecting the United States. In fact, airlines had already canceled the great majority of flights from China, and other countries cut off travel from China at around the same time Mr. Trump did.

The first known case in the United States was confirmed on Jan. 20, after a man who was infected but not yet sick traveled five days earlier from Wuhan to the Seattle area, where the first serious American outbreak would occur.

The W.H.O. said repeatedly that it did not endorse international travel bans, which it said are ineffectual and can do serious economic harm, but it did not specifically criticize the United States, China or other countries that took that step.

Experts say it was China’s internal travel restrictions, more severe than those in the West, that had the greatest effect, delaying the epidemic’s spread by weeks and allowing China’s government to get ahead of the outbreak.

The W.H.O. later conceded that China had done the right thing. Brutal as they were, China’s tactics apparently worked. Some cities were allowed to reopen in March, and Wuhan did on April 8.

The Trump administration has not been alone in criticizing the W.H.O. Some public health experts and officials of other countries, including Japan’s finance minister, have also said the organization was too deferential to China.

The W.H.O. has altered some of its guidance over time — a predictable complication in dealing with a new pathogen, but one that has spurred criticism. But at times, the agency also gave what appeared to be conflicting messages, leading to confusion.

In late February, before the situation in Italy had turned from worrisome to catastrophic, Prime Minister Giuseppe Conte and other government officials, citing W.H.O. recommendations, said the regional governments of Lombardy and Veneto were doing excessive testing.

“We have more people infected because we made more swabs,” Mr. Conte said.

In fact, the W.H.O. had not said to limit testing, though it had said some testing was a higher priority. It was — and still is — calling for more testing in the context of tracing and checking people who had been in contact with infected patients, but few Western countries have done extensive contact tracing.

But the organization took pains not to criticize individual countries — including those that did insufficient testing.

On March 16, Dr. Tedros wrote on Twitter, “We have a simple message for all countries: test, test, test.” Three days later, a W.H.O. spokeswoman said that there was “no ‘one size fits all’ with testing,” and that “each country should consider its strategy based on the evolution of the outbreak.”

The organization was criticized for not initially calling the contagion a pandemic, meaning an epidemic spanning the globe. The term has no official significance within the W.H.O., and officials insisted that using it would not change anything, but Dr. Tedros began to do so on March 11, explaining that he made the change to draw attention because too many countries were not taking the group’s warnings seriously enough.