British regulators are considering two vaccines for emergency approval: one developed by AstraZeneca and the University of Oxford, and another by Pfizer and BioNTech. Prime Minister Boris Johnson visited a vaccine manufacturing facility in Wales on Monday, boasting that “if we’re lucky, if everything goes, right, this could be available just in a few weeks.”

“This could, and I stress could, really be the salvation for humanity, these vaccines,” Mr. Johnson said.

Across the Atlantic, Mark Meadows, President Trump’s chief of staff, summoned the F.D.A. commissioner, Stephen Hahn, to the White House on Tuesday morning to explain why regulators have not yet approved the emergency use of the vaccine developed by Pfizer, according to people familiar with the situation.

One official said Mr. Meadows was concerned that the British government may approve emergency use of a coronavirus vaccine before American regulators act, potentially embarrassing the Trump administration. Mr. Trump and Mr. Meadows have pressured the F.D.A. for months to speed up the development and approval of a vaccine, and allies of Dr. Hahn are worried that he could be fired if he fails to satisfy the White House demands.

On Tuesday, the White House said that Mr. Trump will host a “Covid-19 Vaccine Summit” next week, two days before a panel of outside advisers to the F.D.A. meets to decide whether the agency should grant emergency approval to the Pfizer vaccine. Vice President Mike Pence, the head of the White House coronavirus task force, will also attend the meeting with private sector executives and governors.

The global race to develop a vaccine is poised to shatter records for time to market. Around the world, researchers are testing 57 vaccines in clinical trials, and nearly 100 others are being tested in animals or cells. China and Russia have approved vaccines without waiting for the results of late-stage trials, known as Phase 3, which experts say raises serious risks.

Britain and the United States vet vaccine candidates differently: American regulators pore over raw data from vaccine makers to validate their results, while regulators in Britain and elsewhere in Europe lean more heavily on companies’ own analyses.

The European Union is also set to conduct its own review of the Pfizer and BioNTech vaccine. The companies announced on Tuesday that they had submitted their final application to the bloc’s health authorities. Moderna announced on Monday that it had also applied for approval for the European market and in the United States.

But creating a vaccine is only the first step in what will be a gargantuan feat of logistics: injecting enough people around the world to stop a pandemic in which more than 63 million people have been infected and 1.4 million have died. The United States, Britain and other rich countries have bought massive quantities of doses, but will still have to figure out how to ship, store and distribute them.

The Pfizer vaccine brings particular challenges: It must be stored in an ultracold freezer. That quandary already appeared to be affecting distribution plans in Britain. Nursing-home residents were supposed to be Britain’s top priority under an advisory committee’s plans, but a limit on how many times officials believe the Pfizer vaccine can be moved before it loses effectiveness likely means that National Health Service staff members will receive the shots first.

In the United States, states largely devise plans for vaccine distribution, but are likely to follow recommendations from the Centers for Disease Control and Prevention. An advisory committee was meeting on Tuesday to discuss those.

Governments and health care organizations must also figure out how to overcome mistrust and concern about the vaccine’s safety. On Monday, the head of the International Federation of Red Cross and Red Crescent Societies, Francesco Rocca, warned that misinformation and distrust could torpedo vaccination campaigns.

“To beat this pandemic, we also have to defeat the parallel pandemic of distrust,” he said.