5-6 minutes - Source: BBC
Mr Biden said co-ordination was needed to tackle the coronavirus outbreak.
He called President Trump's refusal to acknowledge he lost the election, despite calls to do so from both sides, "totally irresponsible".
The Trump campaign launched a flurry of legal challenges in the wake of the 3 November vote to contest ballot counts.
The president's team is trying to have courts overturn votes in key states on the grounds that many ballots were invalid or improperly counted. So far those efforts have failed and no evidence of significant fraud has emerged.
President-elect Biden, a Democrat, has 306 votes in the electoral college, surpassing the 270 threshold needed to win.
Yet Mr Trump, a Republican, tweeted on Monday: "I won the Election!"
The government agency that launches transition process - the General Services Administration (GSA), headed by a Trump appointee - has yet to recognise Mr Biden and his running mate Kamala Harris as winners.
This leaves them without access to sensitive government briefings that are normally provided to an incoming administration.
Aides to the president-elect have said that Mr Trump's refusal to engage in a transition also means Mr Biden's team has been excluded from planning around a vaccination distribution strategy.
Speaking in his home state of Delaware on Monday, Mr Biden said of the stalled transition: "Does anyone understand this? It's about saving lives, for real, this is not hyperbole."
"More people may die if we don't co-ordinate," he added.
Calling nationwide vaccine distribution a "huge undertaking", Mr Biden said that if his team had to wait until 20 January - his presidential inauguration - until they could begin work on the distribution programme, they would be behind by "over a month, month and a half".
Asked if he would encourage state leaders to reinstate stay-at-home orders, the president-elect sidestepped, and instead called on officials to encourage mask-wearing.
One White House, different views
National Security Adviser Robert O'Brien spoke of a "professional transition" to the next administration on Monday. His tone was different from that of Secretary of State Mike Pompeo, who spoke of a "second Trump administration" last week, and others who act as though the president will remain.
When I asked Judd Deere, a spokesman here at the White House, about Mr O'Brien's remarks and his own role in the transition, Mr Deere shot the idea down: "There's not a transition at this point." When I asked about advice he'd give to those who will come after him, he got a bit testy: "I don't speak in hypotheticals."
His remarks reinforce those of the secretary of state, but clash with those of the national security adviser, an unsurprising development, given that their boss is himself a bundle of contradictions. He has tweeted about a Biden victory one moment and then reversed himself, and his cabinet secretary, adviser and others reflect his zig-zaggy approach.
How are Trump's challenges going?
The Wisconsin Elections Commission on Monday said the Trump campaign would have to pay nearly $8m (£6m) for a vote recount in that state - which Mr Biden appears to have won by 20,000 ballots - if it still wanted one.
But in a consolation for Mr Trump, Georgia officials said they had found nearly 2,600 ballots that had not been counted in a Republican-leaning county on election night. A voting official in the state, Gabriel Sterling, said the ballots were overlooked because of "a person not executing their job properly".
The isolated discovery was expected to improve Mr Trump's vote count by a net 800 - not by enough to overturn Mr Biden's lead of more than 14,000 ballots.
Georgia is conducting a statewide, by-hand recount because of the 0.3% margin separating the rivals.
Meanwhile, the official overseeing the recount, Georgia Secretary of State Brad Raffensperger, told CNN on Monday that he had been coming under pressure from a fellow Republican, Trump ally and South Carolina Senator Lindsey Graham, to disqualify legally posted ballots in certain counties.
Mr Graham denied the claim, telling Politico the two had merely had a "very pleasant" conversations about signature verification processes.
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