BRUSSELS — The people on television were joyous: Jubilant Britons were receiving the world’s first shots of the Pfizer-BioNTech vaccine in early December.Less joyous were many people watching in Germany, where the vaccine was created yet where the government was telling citizens it would be weeks before they could launch their own vaccination program.“Millions Getting German Vaccine, but We Have to Keep Waiting,” read the headline in the Berlin tabloid B.Z. “The World Is Vaccinating — Not Germany,” read the newsmagazine Focus.For Germans and other Europeans, it has been particularly galling to watch as the United States and Britain, which were less disciplined in their lockdowns and pandemic precautions, have vaulted ahead in the vaccine race. In fact, former President Donald J. Trump and Prime Minister Boris Johnson had all the more incentive to grasp at vaccines as their countries became among the worst hit in the world.There is no doubt that the European Union bungled many of the early steps to line up vaccines. It was slower off the mark, overly focused on prices while the United States and Britain made dollars and pounds no object, and it succumbed to an abundance of regulatory caution. All those things have left the bloc flat-footed as drugmakers fall behind on their promised orders.But the 27 countries of the European Union are also attempting something they have never tried before and have broken yet another barrier in their deeper integration — albeit shakily — by choosing to cast their lot together in the vaccine hunt.In doing so, they have inverted the usual power equation of the bloc. Bigger, richer countries like Germany and France — which could have afforded to sign contracts directly with drugmakers, as the United States and Britain did — saw their vaccine campaigns delayed by the more cumbersome joint effort, while smaller countries wound up with better supply terms than they were likely to have negotiated on their own.ImageVaccinations in Cardiff, Wales, on Dec. 8, the first day of the British program, which has already reached 16 percent of the population.Credit...Andrew Testa for The New York TimesFor the bulk of E.U. nations, that experiment has been beneficial. But it has not necessarily been greeted happily in the disadvantaged wealthiest countries, and it has left leaders like Chancellor Angela Merkel of Germany and President Emmanuel Macron of France open to criticism at home.They and E.U. leaders have nonetheless stood by their decision and the impulse for solidarity, even as the finger-pointing has begun.“What would people have said if Germany and France had been in competition with one another for the purchase or production of vaccines? That would have been chaos,” Mr. Macron told a news conference on Friday, after a virtual meeting with Ms. Merkel. “That would have been counterproductive, economically and from a public health perspective, because we will only come out of this pandemic when we have vaccinated enough peo