The mayors of London, Greater Manchester and other English cities have written to the government urging faster implementation of “desperately needed” changes to gender recognition legislation.

Sadiq Khan, London’s mayor, wrote to the equalities minister, Penny Mordaunt, along with Andy Burnham, the mayor of Greater Manchester, and Steve Rotheram and Dan Jarvis, the mayors of the Liverpool and Sheffield city regions.

The Labour politicians encouraged Mordaunt to introduce the changes soon to make it easier for trans people to have their gender identity legally recognised. They argue that the Gender Recognition Act (GRA) 2004 is “now outdated and in urgent need of reform”.

Campaigners have criticised the current process – whereby trans people should supply evidence of a medical diagnosis of gender dysphoria and of having lived in their “acquired gender” for two years – as humiliating and bureaucratic.

The government estimates that 12% of trans people have a gender recognition certificate. In a national LGBT survey last year, 92% said they would be interested in getting one.

The mayors argued: “GRA reform is a key step in addressing the marginalisation of trans and non-binary communities by allowing them to more easily gain legal recognition of their gender identity.”

They said the current GRA process “creates unnecessary barriers” preventing trans people from achieving legal recognition. “The government has made commitments to ‘streamline and demedicalise’ the gender recognition process, a promise that was welcomed by trans and non-binary people across the country. We hope to see this reflected in the new legislation,” they told Mordaunt.

The proposed changes have stirred great debate, particularly about who can access single-sex services such as domestic violence refuges. Although the changes would not alter exceptions that make provision for single- and separate-sex spaces, some groups argue that self-ID is potentially open to abuse by predatory men seeking to access women’s spaces and women’s bodies. The consultation closed in October last year.

A number of countries have already made it easier for trans people to be recognised in their chosen gender identity. The mayors said in their letter to Mordaunt: “Changing the law would bring us in line with the standard already set by Ireland, Denmark and Norway, whilst transforming the lives of trans and non-binary people.”

Scotland has postponed similar changes to its gender recognition laws after criticism it had not addressed concerns about the impact of its proposals. It sparked public tensions within the Scottish National party and a fierce debate among politicians, campaigners and women’s groups which Scotland’s equalities secretary, Shirley-Anne Somerville, labelled as “toxic”.

Paul Martin, the chief executive of the LGBT Foundation and a government adviser on LGBT issues, welcomed the mayoral intervention. He said: “We are delighted to see that four mayors from some of our country’s biggest metropolitan areas have joined together to make such a strong statement of support for their trans and non-binary constituents.

“In these turbulent political times it can be easy to forget that the everyday business of government must continue. The Gender Recognition Act is outdated and is in need of urgent reform to meet the needs of the communities it was originally designed to support.

“We look forward to the upcoming announcement and call on the government to allow trans people to legally change their gender by self-declaration, introduce legal recognition for non-binary people and improve inclusion for young trans people.”