Entire families gathered in the streets of for news of the deal brokered by Russia and Turkey for their safe passage.
The delicate deal to end years of rebel resistance in Syria's second city came amid mounting international concern about the plight of civilians and as the UN said it had reports of atrocities being committed by advancing government forces.
Russian Foreign Minister Sergei Lavrov said Wednesday that Moscow expects rebel resistance in Aleppo to end in the next two to three days.
"I expect that the rebels will stop their resistance in the next two to three days," Russian news agencies quoted Lavrov as saying. "I hope that the situation in eastern Aleppo will be resolved in two to three days."
From before dawn, entire families gathered in the streets of for news of the deal brokered by Russia and Turkey for their safe passage.
The first departures had been expected to begin around 5:00 am (0300 GMT), but hours later there was no sign of movement.
There was no comment on the apparent delay in the evacuation from the government or the rebels, or from the deal's co-sponsors.
An AFP correspondent in the government-held west of the city said evacuation buses waiting on the the edge of the Salaheddin neighbourhood -- still divided between rebel and regime control -- had not moved.
Drivers were sleeping inside the two dozen green government buses, which arrived late on Tuesday soon after the deal was announced.
An AFP correspondent inside remaining rebel-held territory said hundreds of civilians had gathered in the streets from the early hours, some clutching bags of belongings, to await evacuation.
Some had slept in the open, despite the cold and a fierce storm that brought heavy rain and high winds, after fleeing from other districts as they army advanced and finding nowhere to shelter.
No aid has entered east Aleppo since mid-July, when government troops first encircled rebel districts of the city, and food supplies have dwindled, forcing many to survive on as little as a meal every two days.
The army has seized virtually all of east Aleppo, in rebel hands since 2012, since beginning an operation to recapture the city last month.
Losing their onetime bastion will deal the opposition its worst blow since the conflict began in March 2011 and mark a major victory for President Bashar al-Assad and his allies Iran and Russia.
The evacuation deal emerged late Tuesday, with rebel officials saying civilians and fighters would leave for opposition territory elsewhere in northern Syria.
"The first stage will be the evacuation of civilians and wounded, within hours, and afterwards fighters will leave with their light weapons," said Yasser al-Youssef, of the Nureddin al-Zenki group.
Turkey said those leaving would be taken to Idlib, which is controlled by a powerful rebel alliance that includes Al-Qaeda's former affiliate Fateh al-Sham Front.
Deputy Prime Minister Mehmet Simsek said Turkey would "set up a tent city to accommodate up to 80,000 Syrian refugees fleeing Aleppo".
Washington, which backs the opposition, said it had not been consulted on the deal but welcomed any arrangement that brought safety to the people of Aleppo.
US ambassador to the United Nations Samantha Power urged international supervision of the evacuation, expressing concern about "people who wish to leave but who, justifiably, fear that, if they try, they will be shot in the street or carted off to one of Assad's gulags."
While tens of thousands of civilians have already fled east Aleppo for government-held territory, others say they fear arrest or torture at the hands of the regime.
The UN has reported allegations that men have gone missing after crossing to government areas, and on Tuesday said it had credible reports of summary executions by pro-regime fighters.
UN rights office spokesman Rupert Colville described reports of at least 82 civilians, including 11 women and 13 children, being executed in recent days.
He said pro-government fighters had reportedly entered homes and killed those inside, and in others "caught and killed on the spot" fleeing civilians.
The UN was "filled with the deepest foreboding for those who remain in this last hellish corner of opposition-held eastern Aleppo," he said.
Aleppo, a cultural and economic hub second only to Damascus in importance, had been split between a rebel-controlled east and government-held west since 2012.
It was unclear how many civilians remained in rebel territory, after an estimated 130,000 fled to other parts of Aleppo during the government advance.
The assault has been accompanied by heavy air and artillery bombardment and at least 463 civilians, including 62 children, have been killed in east Aleppo since it began in mid-November, according to the Syrian Observatory for Human Rights.
Another 130 people, including 40 children, have been killed by rebel rocket fire on western districts, the monitoring group says.
More than 310,000 people have been killed since the Syrian conflict began, and over half the population has been displaced, with millions becoming refugees.