When a bank is on the verge of collapse, financiers and politicians are apt to rally round, saying it is "too big to fail". The leading UK charity Refugee and Migrant Justice, which is under threat of closure, may not command such attention, but to the thousands of asylum seekers it supports each year, it is an essential lifeline.

This week, leading figures including the Archbishop of Canterbury joined calls to save RMJ, which is the largest charity of its kind in the UK and currently provides legal advice to around 10,000 vulnerable people, including 900 unaccompanied children. It is threatened by changes to the legal aid system that mean money is now paid upon completion of case work.

As a result, the charity faces a huge cash shortfall. Caroline Slocock, RMJ's chief executive, said:

RMJ is not asking for new money, simply prompt payment of legal aid for the work it has carried out. Until recently we were given regular, ongoing payments for the work we do. As a not-for-profit organisation we cannot expect to make sufficient profit to finance millions of pounds of working capital.

In its defence, the Minstry of Justice says that RMJ has had three years notice of the legal aid changes and that it has worked with the charity to help minimise "disruption". But a spokesperson for the charity pointed out that while it needs £1.8m over the next six months to survive, the closure of RMJ would cost the government more than £2m as the government would need to pay other providers to take on cases.

The issue is about more than money, however. No-one knows the value of RMJ's work better than John "Bosco" Nyombi, who fled Uganda in 2001 in fear for his life. Nyombi, 31, is gay and his lifestyle had long made him the target of abuse in his home country, where homosexuality is a crime.

The final straw came when police raided a bar near his home. "My friends were arrested and they were beaten up really severely," he says. "The police mentioned my name, they went to my mum's house and my place of work. That's when I decided to leave."

If Britain honoured fully its commitment to the 1951 UN Refugee Convention, which defines a refugee as someone with "a well-founded fear of being persecuted for reasons of race, religion, nationality, membership of a particular social group or political opinion," Nyombi's struggle might have ended there. But instead, he fought a six-year battle with the UK Home Office to have his asylum claim accepted.

In 2008, he was detained for four months ahead of being deported back to Uganda, during which time he was not allowed to contact his solicitor. But lawyers at RMJ working on his behalf were able to appeal against the decision and Nyombi was finally granted leave to stay.

"I wouldn't be here talking now if it wasn't for RMJ," he says, simply.

The uphill struggle that asylum seekers face in this country is a particularly poisonous legacy of the last government. Thanks to Labour's attempts to seem "tough" on immigration, refugees who arrive in the UK are confronted by an inhumane system where adults and children alike are detained for months on end despite having committed no crime. The asylum system is under constant pressure from strict government targets and there have been allegations of institutional racism at the UK's main asylum processing centre in Wales .

The new administration has already made one positive step on asylum in pledging to end child detention. Its wider attitude to immigration remains unclear, with one half of the coalition having pledged limits and the other having favoured an amnesty for illegal immigrants, prior to the election.

Swift action to help RMJ would indicate how far Cameron's government is going to live up to its claim to progressive politics - a failure to do so will reveal just how little substance lies behind the "Big Society" window-dressing.