The number of people fleeing Russia’s advance into Ukraine has reached two million, the UN refugee chief said on Tuesday.

Refugees wait in freezing cold temperatures to get on a bus after crossing the Ukrainian border into Poland (AFP via Getty Images)© Provided by Evening Standard Refugees wait in freezing cold temperatures to get on a bus after crossing the Ukrainian border into Poland (AFP via Getty Images)The grim total, announced by United Nations High Commissioner for Refugees Commissioner Filippo Grandi, was reached as buses packed with people fleeing the Russian invasion began a procession out of the eastern city of Sumy.

But while many flee, others are trapped inside besieged cities that are running low on food, water and medicine amid the biggest ground war in Europe since World War II.

Previous attempts to lead civilians to safety have crumbled with renewed attacks.

The route people took on Tuesday out of the eastern city of Sumy was one of five promised by the Russians to offer civilians a way to escape the fighting.

Video posted by the Ukrainian state communications agency showed people with bags boarding buses, but it was not clear how long the effort would last.

“The Ukrainian city of Sumy was given a green corridor, the first stage of evacuation began,” the agency tweeted. Sumy is just 50 kilometers (30 miles) from the Russian border.

Ukrainian Deputy Prime Minister Iryna Vereshchuk said Tuesday that both sides agreed to a ceasefire from 9am-9pm Ukraine time (7am-7pm GMT) for the evacuation of civilians from Sumy.

The first convoy with evacuated civilians in buses or private cars left on a single route toward the Ukrainian city of Poltava.

Those being evacuated from Sumy include foreign students from India and China, she said. The corridor will also be used to bring humanitarian aid into Sumy, she added.

It comes as the UK faces growing criticism over its efforts to provide refuge to Ukrainian refugees.

The Home Office disclosed on Monday night that just 300 visas have been issued out of a total of 17,700 family scheme applications that have been started, 8,900 of which have been submitted.

Defence Secretary Ben Wallace acknowledged that the Government needs to move quicker and said he is offering Ministry of Defence (MoD) support to speed up the work.

“The first and foremost duty for all of us is to make sure that people get to safety,” he told Sky News.

“Once they’ve got to safety, making sure we just check their identity before they come to this country – it is incredibly important that we do that.

“It shouldn’t take time. And I’ve offered, I will be offering, to the Home Office assistance from the MoD in the same way we did in Op Pitting (the evacuation of Afghanistan) to increase the processing time to help those people.”

He added: “Of course, we can do that quicker, we are leaning into that, the Home Secretary is determined to do that quicker, I will give her all the support I can.”

Mr Wallace was also critical of reports that hundreds of Ukrainians who have reached Calais have been told they need to travel back to Paris or Brussels to apply for a visa where they face a lengthy wait.

“It’s difficult for those people – why wouldn’t it be? – to go all the way back to Paris,” he told BBC Breakfast.

“We can do more, we will do more.”

Two million refugees have now fled Russia’s war in Ukraine as residents evacuate eastern city of Sumy (msn.com)