UN High Commissioner Filippo Grandi said the burden on European countries hosting the refugees would be greater, as the agency reported two million people have now left Ukraine.
His comments come as Prime Minister Boris Johnson insisted the UK would be as generous as possible to Ukrainians fleeing the Russian invasion despite criticism over the slow pace of processing visa applications.
Meanwhile, a British-Ukrainian man who has spent several days in Calais trying to help his family flee their war-torn homeland described his frustration at the red tape and delays which have halted their safe passage to the UK.
Civilians in Ukraine have begun to evacuate towns which are under the greatest levels of attack, including Sumy, in the north east of the country, and the town of Irpin, outside the capital Kyiv, in third attempt to create humanitarian corridors for safe passage.
UK defence secretary Ben Wallace offered military support to the Home Office to help deal with the backlog of people trying to get into the UK.
Meanwhile, foreign secretary Liz Truss announced a new visa processing centre is being set up in Lille in northern France following reports that refugees arriving in Calais were being told they must go to Paris or Brussels to apply.
The Prime Minister briefed the weekly meeting of the Cabinet on the latest situation in Ukraine ahead of a historic address to the House of Commons by President Volodymyr Zelensky on Tuesday evening.
A No. 10 spokesman said Mr Johnson told Cabinet the UK was continuing to supply "lethal defensive aid and equipment" to bolster the Ukrainian forces' "heroic" resistance in the face of the Russian onslaught.
The Prime Minister said the UK would be "as generous as we could" in its support for Ukrainian refugees, while home secretary Priti Patel set out the extra assistance the UK Government is providing.
The Home Office disclosed on Monday night just 300 visas had been issued out of a total of 17,700 family scheme applications that had been started – 8,900 of which have been submitted.
Senior Tory backbencher Sir Geoffrey Clifton-Brown, who secured an emergency Commons statement, said while security checks were important, it should be possible to speed up the process.
"We need efficiency, but also humanity when it when processing applications of refugees from Ukraine, and we should warmly welcome those refugees to this country," he said.
Mr Zelensky separately called for the expansion of humanitarian corridors for civilians fleeing the war, and for more support from the Red Cross.
In a video address from an undisclosed location, Mr Zelensky said a child had died of dehydration in the blockaded southern port of Mariupol, in a sign of how desperate the city's population had become.
Authorities said buses had carried civilians out of one embattled Ukrainian city and supplies towards another. However, reports of renewed Russian attacks in one area threatened to again derail those efforts on the 13th day of the invasion.
Transport left from the north-eastern city of Sumy, according to video from the Ukrainian communications agency. Deputy prime minister Iryna Vereshchuk said they were heading south west to the city of Poltava, and included students from India and China.
Buses emblazoned with red cross symbols carrying water, basic food staples and medicines moved towards the besieged southern port of Mariupol, to ferry civilians out of the city, but the Ukrainian president's office later said it had been informed of shelling on the escape route.
Human Rights Watch (HRW) said “repeated” attacks on a supposed corridor two days ago suggested Russian forces had violated their obligations under international humanitarian law.
The agency said: “For several hours on March 6, 2022, Russian forces bombarded an intersection on a road that hundreds of civilians were using to flee the Russian army’s advance in northern Ukraine to Kyiv.
"The repeated nature of the attacks, which according to the government killed at least eight civilians, suggests that Russian forces violated their obligations under international humanitarian law not to conduct indiscriminate or disproportionate attacks that harm civilians, and failed to take all feasible measures to avoid civilian casualties.”
Mr Grandi said the first waves of refugees seen coming out of Ukraine were those with "some resources".
"Many come by car, and especially they have connections,” he said. “They can go where they have family, friends, communities.”
However, Mr Grandi said more people leaving Ukraine would be those in less comfortable financial circumstances, who would need more aid.
"That will be a more complex issue for European countries to manage going forward and there will need to be even more solidarity by everybody in Europe and beyond," he said.
Sales manager Vitalii Morgun, 36, has spent the past week-and-a-half in continental Europe supporting his family's quest for refugee status in the UK after Russian forces waged war on Ukraine.
Mr Morgun, who has lived in the UK for almost two decades, said his brother Yevgen, sister-in-law Anna, and their two young children have been effectively stranded in the French port city of Calais after assurances their visas would be granted by Friday last week did not come to fruition.
They are living in a hotel, but say the money is running out and fully expect the lack of progress from the UK Government to continue for another week.
Mr Morgun said: "We've tried to call all the possible offices and departments and visa centres and no-one knows anything. They just keep pointing the finger at each other and say we have to wait."
Mr Morgun, who lives in Romford, east London, said the family submitted all the paperwork at a visa centre in Brussels on Thursday last week, having spent five days travelling from Kharkiv to the Hungarian border where they met Mr Morgun and his father, Volodymyr.
Mr Morgun said his mother, who is in the UK, cannot stop crying due to the fear of what will happen to the family.
He said: "Our town [Kharkiv] has been almost completely destroyed. We don't know what the situation is there. There is absolutely no chance for them to go back to that town in the near future. They have nowhere else to go, so I'm trying to get them to join me in the UK."
Mr Morgun added: "There is an option for them [family members] to join me as refugees, but no-one knows how the programme works at the moment. It's been very difficult. I've got somewhere for them to live, I have the paperwork. We just need to get them across the border.
"I know they have a lot of bureaucracy in the UK and things are taking ages. I wish it could be better, because I really love the UK. But it's just the way things are."
He said he had lots of Ukrainian friends in similar situations, none of whom had yet been granted visas.