The UK's biggest ever inoculation programme is due to start on Monday.
The FDA has been reviewing thousands of pages of technical information on the Pfizer and Moderna vaccines as it deliberated over authorising "emergency use" for high-risk groups.
It scheduled public meetings for December 10 and December 17 to review the vaccines, and was expected to make a decision before January.
Pfizer applied for emergency use authorisation on November 20.
White House officials frustrated by the time lag were reportedly unhappy that Hahn spent a week in North Carolina last month. The FDA responded that he was working remotely from there while quarantining after exposure to the virus.
Trump has told aides he believes a slow approval process for the vaccine was continuing to undermine his popularity.
Despite the pressure from the White House, Hahn said the FDA would take the time needed to "get this right".
"No one at FDA is sitting on his or her hands. Everyone is working really hard to look at these applications and get this done. But we absolutely have to do this the right way.
"We will make sure that our scientists take the time they need to make an appropriate decision. It is our job to get this right and make the correct decision regarding vaccine safety and efficacy."
There were also differences between the UK and US approval processes.
Hahn said: "FDA is one of the few regulatory agencies in the world that actually looks at the raw data. We are going to do our own analysis."
The European Union is waiting for a review and go-ahead from the European Medicines Agency. Britain, which is still under EU rules, is using a loophole in EU law that allows countries to put a vaccine out in an emergency rather than through the EMA process.
"The idea is not that we be first, but ... to have safe and effective vaccines in the pandemic," German Health Minister Jens Spahn said.
Kayleigh McEnany, the White House press secretary, said: "We will never apologise for putting the fire under these agencies. We want a safe vaccine, absolutely. We also want a fast one because lives are at stake and a vaccine by the end of the year is key and paramount."
Asked if the White House was given advanced word about the UK vaccine approval, McEnany said: "I'm not aware of any heads-up that we were given."
Trump believes that the vaccine should be a key part of his presidential legacy.
Last weekend, he said: "They will try and say that Biden came up with the vaccines."