By Mohamed Osman in Khartoum and Oliver Slow in London
The situation was now “untenable” for civilians left without food or water, and some hospitals had stopped working, spokeswoman Alyona Synenko said.
Convoys leaving the capital Khartoum had encountered robbery and looting, US Secretary of State Antony Blinken said.
People getting out of Khartoum spoke of corpses left lying on the street.
“We saw bodies everywhere – there’s no security whatsoever so nobody dares go collect them – but there’s utter destruction too. Everything is just devastated,” Italian NGO boss Stefano Rebora told the BBC.
South African diplomat Clayson Monyela said all routes out of Khartoum – a city of six million people – were “risky and dangerous”.
“The airport remains closed, the fighting continues,” he told the BBC. He reiterated a call for a ceasefire to allow people to leave, and aid to enter.
Sudan was suffering an “internet blackout” with connectivity at 2% of ordinary levels, monitoring group NetBlocks said on Monday. In Khartoum, the internet has been down since Sunday night.
Meanwhile, water and electricity have been restored to some parts of the capital, but not all of them.
One Nigerian student told the BBC: “The taps are outside in the street and that’s the scary part – sometimes we are going out to get water but there is shooting or explosions going on, so we just have to run there and get the water and come back.”
Numerous countries have evacuated their civilians – and thousands of other people have made risky escapes.
Many of those who have left Khartoum have headed to other parts of the country where they have family ties – leaving parts of the city centre completely deserted.
Others have gone north to Egypt by bus, or headed south.
Officials in neighbouring South Sudan said the roughly 10,000 refugees who had arrived in recent days came from Eritrea, Kenya and Uganda – as well as from Sudan and South Sudan themselves.
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Multiple countries have stepped up efforts to evacuate diplomats and civilians from Khartoum.
By Monday about 1,100 European Union citizens had been taken out of Sudan, an EU diplomatic source told the BBC. The bloc believed about 1,700 EU citizens had been in Sudan when the fighting began.
The US said it had airlifted fewer than 100 people by helicopter on Sunday in a “fast and clean” operation. The American embassy in Khartoum is now closed, and a tweet on its official feed says it is not safe for the government to evacuate private US citizens.
The UK government airlifted British diplomats and their families out of the country. Foreign Minister James Cleverly said options to evacuate remaining Britons were “severely limited”. Canada has evacuated its diplomatic staff.
Turkey – a key player in Sudan – began evacuation efforts by road from the southern city of Wad Medani on Sunday, but plans from one site in Khartoum were postponed after a nearby explosion.
More than 150 people – mostly citizens of Gulf countries, as well as Egypt, Pakistan and Canada – were evacuated by sea to Saudi Arabia.
Long lines of UN vehicles and buses were seen leaving Khartoum on Sunday, heading east towards Port Sudan on the Red Sea and carrying “citizens from all over the world”, a Sierra Leonean evacuee told AFP news agency.
However many foreign students from Africa, Asia and the Middle East are among the foreigners still trapped in Khartoum.
The western region of Darfur – where the RSF first emerged – has also been badly affected by the fighting.
The UN has warned that up to 20,000 people – mostly women and children – have fled Sudan to seek safety in Chad, across the border from Darfur.
However in other parts of the country, some semblance of normalcy has emerged.
In Omdurman, across the Nile from Khartoum, there has been less gunfire and explosions than in previous days, possibly to allow civilians to leave. Heavy fighting outside the army headquarters has stopped.
As a result, for the first time since hostilities broke out more than a week ago, women and children have been out on the streets, visiting neighbours and going to markets, which still have some basic supplies such as oil and wheat. There are long queues outside the few bakeries that remain open.
More than 400 people have died in the conflict, and thousands have been injured, according to the latest tally from the World Health Organization. But it is feared the true toll is much higher.
Several ceasefires that seemed to have been agreed were subsequently ignored – including a three-day pause to mark the Muslim holiday of Eid al-Fitr, which started on Friday.
The UN’s World Food Programme says the fighting could plunge millions more Sudanese into hunger in a country where a third of the population already struggles to get enough to eat.