Police have been accused of failing to protect a hate crime-fighting group that was championed by Jo Cox, the MP who was shot and stabbed to death in June.

Tell MAMA, which helps victims of Islamophobia, says it has suffered a campaign of abuse that has peaked in the last eight days, with a barrage of hate messages blocking its victim helpline.

The group says it has repeatedly asked the Metropolitan police for help and was amazed to be told it was considered to be at low risk of harm. It says the campaign of harassment from people spouting racist abuse has paralysed the organisation.

It claims that while police have listened to recordings of the hate messages they have not secured them as evidence.

On Tuesday the government announced new measures to tackle hate crimes, including a review of how police handle them.

Police in England and Wales say hate crime went through the biggest-ever recorded increase during and after the EU referendum campaign.

Tell MAMA says abuse spiked eight days ago and that there has been a concerted campaign of abuse on its helpline, with hate calls coming in quick succession.

Fiyaz Mughal, founder and director of Tell MAMA, told the Guardian: “In the last eight days, we have had to disconnect our phone lines and have not been able to service victims of anti-Muslim hatred because of the incessant abusive calls we have received and which have intimidated my female staff. The harassment has been so bad that we have had to leave the phones offline.”

Mughal said the abuse included racist and misogynistic terms and anti-Muslim abuse.

Mughal added: “A national hate crime project under constant abuse and focus by anti-Muslim bigots and far-right extremists clearly fits into a project that has risks. We are in a complex and fast-changing environment where my team should not have to put up with a poor response from the Met and with assessments of risk that have no relevance to the actual risk.”

Mughal said harassment had been suffered for years and the Met’s response had been similarly poor, in his judgment. “For over four years, my colleagues and I have been maliciously targeted with threats, floods of malicious calls, viruses being sent to us, death threats and general anti-Muslim bigotry.

“We have approached the Met and spent tens of hours filling in paperwork with little or no action taken by them. The Met have not brought a single person to book for the targeting of us.”

The Met said: “We are currently investigating this allegation that has been reported in the last two weeks and are engaging with Tell MAMA about how we can assist them.”

Tell MAMA was supported by Cox, who was killed by a man whose case will be treated as terrorist as it is handled by the court system.

Mughal said: “After Jo Cox, we are very on edge. Does it take someone to be physically injured for the police to get their act together?”

The home secretary, Amber Rudd, said a new package of measures to boost the fight against hate crime would involve Her Majesty’s Inspectorate of Constabulary examining the police response.

Rudd said: “This government is determined to build a Britain that works for everyone. Those who practise hatred send out a message that it’s OK to abuse and attack others because of their nationality, ethnicity or religious background; that it’s OK to disregard our shared values and promote the intolerance that causes enormous harm to communities and individuals.”

Amid concern over the rise in hate crime since the vote for Brexit, one of Britain’s most senior judges, Lord Dyson, warned on Tuesday that tolerance in Britain was “being put under strain by xenophobia and forces of hate”.

Dyson, the master of the rolls, was speaking at a ceremony to mark his retirement. “I feel a great debt of gratitude to this country for allowing my forebears to settle here,” he said. Dyson’s mother came from Bulgaria and his father’s parents came from Lithuania.

“I fervently hope that the events of recent weeks have not put that tolerance at risk. I am fearful that it is being put under strain by xenophobia and forces of hate.”

He added: “I do have faith in the British people. I believe their tolerance will continue to shine.”