No reason was given for the delay. The International Red Cross said snow has been falling and many are struggling to keep warm.
According to Syrian state TV, a number of buses with armed people and their families were leaving the Al-Sokkari area in eastern Aleppo and heading to Al-Rashdin in western Aleppo.
It said 20 buses departed from eastern Aleppo on Wednesday, while 1,500 civilians are expected to be evacuated from Kafraya and Foua villages in mainly rebel-held Idlib province simultaneously.
Israel offers help
On Tuesday, Israeli Prime minister Benjamin Netanyahu said Israel wanted to help treat those who had been wounded in Aleppo.
Israel has treated thousands of Syrians
over the past couple of years at the Ziv Medical Center in the north of the country.
"We see the tragedy, the terrible suffering of civilians," Netanyahu said.
"I've asked the Foreign Ministry to seek ways to expand our medical assistance to the civilian casualties of the Syrian tragedy, specifically in Aleppo where we're prepared to take in wounded women and children and also men if they're not combatants.
"We'd like to do that, bring them to Israel, take care of them in our hospitals, as we've done with thousands of Syrian civilians.
"We're looking into the ways of doing this, but it's being explored as we speak."
Tens of thousands evacuated from Aleppo
On Tuesday, state-run TV station Al-Ikhbariya reported the Syrian army had used loudspeakers to urge remaining rebels to leave the area, warning it intended to enter the enclave to "remove remnants of terrorism."
More than 37,000 people have already been evacuated from Aleppo, Turkish Foreign Minister Mevlut Cavusoglu tweeted. The remainder, he hoped, would be out of the besieged city by Wednesday.
The International Committee of the Red Cross has put the number of evacuees since last Thursday at 25,000 and said the operation is "still ongoing."
A further 750 people have so far been evacuated from the largely Shiite villages of Fuaa and Kafraya in Idlib province as part of the same deal, the ICRC said.
A shaky ceasefire, a careful evacuation
By the end of last week, the Syrian army had made enough progress in its offensive for President Bashar al-Assad to refer to the progress made as the "liberation" of Aleppo.
"History is being written by every Syrian citizen," Assad said.
After reclaiming some rebel territory, Syrian officials and rebel leaders agreed to a ceasefire designed to grant civilians safe passage out of the city. Humanitarian workers initially helped about 9,000 people leave Aleppo behind.
This past Friday, the evacuation stopped, placing the fragile accord in doubt, after Iranian militias targeted the road that leads into besieged areas of eastern Aleppo with 23-millimeter gunfire, according to the Aleppo Media Center, or AMC, activist group.
Tens of thousands of refugees were left in limbo, many of the them forced to sleep outdoors, or in bombed-out buildings, as temperatures plunged below zero.
Both sides struck another agreement the following day to provide safe passage to those loyal to Assad's regime, including Iranian militia groups, from areas held or besieged by rebels, AMC said. It was a new demand in exchange for the evacuation of civilians, rebels and others loyal to the opposition from eastern Aleppo.
On Monday, the United Nations Security Council voted unanimously to redeploy staff to Aleppo to monitor and report on the evacuation of civilians.
Leaving or left behind in Aleppo?
It's not clear how many civilians are left in Aleppo, but the evacuations could wrap up this week.
Of those leaving, some will head to Idlib province. But for many, being taken to Idlib is trading one war zone for another. Idlib is widely expected to be the regime's next target, and the evacuations are effectively moving rebels to one containable zone.
Turkey is building a new refugee camp for the evacuees from eastern Aleppo. The camp is in the countryside of Idlib and will be able to host 50,000 people when finished.
Turkey, whose once-icy ties with Russia have thawed in recent months, found itself pulled further into the conflict this week as a police officer assassinated the Russian ambassador in Ankara
while shouting: "Do not forget Aleppo!"
Russia and Turkey have agreed to work together on an investigation -- and have vowed to not let the shooting derail progress made toward the end of the war in Syria.