Some $1.5bn (£1bn) in aid pledged to Syria has largely failed to materialise, the UN said on Thursday, as the number of refugees who have fled the country reaches one million.

Gulf countries and other states including the UK pledged the money at a major donor conference in Kuwait last month. About $1bn was earmarked for neighbouring countries hosting refugees, with $500m for humanitarian aid to Syrians displaced internally by fighting.

But with the situation on the ground worsening dramatically since early this year, and with violence reaching unprecedented levels, the UN said it had received little of the $1.5bn promised. The UN's World Health Organisation (WHO) said it had received nothing. The UN refugee agency said it had got 30% of $496m pledged.

Speaking to the Guardian, the head of WHO in Syria, Elizabeth Hoff, said the security situation in the Syrian capital had got worse. There were several explosions on a daily basis, and an increasing number of people in Damascus being killed and injured, she said, adding: "It is much more serious here over the last few weeks than it previously was."

Those hospitals not destroyed in fighting were struggling to cope, with very high occupancy rates. Health workers in Aleppo and the suburbs of Damascus were finding it difficult to get to work because of checkpoints and snipers, she said. Medical supplies – including anaesthetic for trauma surgery – were inadequate. Insulin was running out.

She added: "We have not received any fresh money as of today. And the needs are actually enormous now. The needs are growing: the escalation of the violence, more and more burns. We cannot meet these needs, and we have not received any fresh money."

Last month, Saudi Arabia, Kuwait and Qatar each pledged $300m. But WHO said it was still relying on funds from traditional sources such as Scandinavia, as well as the UN and EU. On Wednesday, Oxfam said just 20% of the $1.5bn pot had reached aid agencies, despite the fact the country's humanitarian crisis was now spinning out of control.

On Thursday the opposition Syrian National Coalition (SNC) said 3.4 million Syrians in the mainly rebel-controlled north needed urgent humanitarian assistance. At least 1 million of them were internally displaced. Little aid was reaching them because Syria's president, Bashar al-Assad, had refused to give aid agencies permission to carry out cross-border humanitarian operations, it said. It called on Russia and China to press Assad to lift the blockade.

According to the latest UN figures, the number of refugees fleeing to neighbouring countries has reached 940,131, a record. The true figure may be higher. Since early January around 40,000 have fled every week – the fastest rate ever. On Wednesday Adrian Edwards, chief spokesman for UNHCR, predicted the number would soon pass the symbolic 1 million, "probably in early March".

Inevitably this exodus has placed a huge strain on Jordan, Lebanon and Turkey, who have complained that they are no longer able to cope. On Thursday, Lebanon's interior minister said Syrian refugees had become a threat to Lebanon's security because of the suspicion that many were rebel fighters.

Residents in northern Lebanon say rebels pose as refugees to cross the border, and are arming members of the refugee community in Lebanon to fight in Syria. The minister, Marwan Charbel, has said Syrian rebels have set up training camps in Lebanon.

In addition, members of the rebel Free Syrian Army have used Lebanon's mountainous terrain to regroup before staging attacks on the Syrian army across the poorly demarcated border. "What is concerning me is the security situation," Charbel said at a joint news conference with the United Nations Development Programme. "Who is exploiting (the Syrian refugees)? Who is arming them? We are not controlling them."