Priti Patel must shoulder some blame for the public’s plunging trust in the police in the wake of Cressida Dick’s resignation, her Labour counterpart said on Sunday.

It comes as Keir Starmer placed the Tories’ record on crime at the heart of Labour’s local election campaign which launched just days after the Met commissioner’s departure, and as Boris Johnson faces possible fines for attending parties during lockdown.

Yvette Cooper said the home secretary had been “silent” on policing for a year while failing to implement reforms or defend the model of policing by consent.

Her criticisms follow Dick’s dramatic resignation on Thursday evening after the mayor of London, Sadiq Khan, made clear he had no confidence in her plans to reform the service.

On Sunday morning, Labour and the Conservatives traded blows over police failings and who was to blame for Dick’s surprise resignation after a series of scandals around misogyny and racism within the Metropolitan police.

Appearing on BBC One’s Sunday Morning programme, Cooper, the shadow home secretary, said Patel had failed to tackle systemic and cultural issues including reforms to training, vetting and misconduct strategy.

“At the moment, there’s been none of those reforms from the home secretary. The home secretary has been silent on policing for a year,” she said.

Sources close to Patel have hit back, claiming the home secretary had begun reforming police training colleges and police crime commissioners, and had sent Her Majesty’s Inspectorate of Constabulary into the Met.

“What does she [Cooper] actually want to do? She just says reform vetting and training, but what?” said an aide.

In the latest strategic move by Labour which will seek to capitalise on the possibility of fines being imposed on Johnson and his staff, Starmer said that the government “do one thing and say another” when it comes to tackling crime.

The former director of public prosecutions said violent youth crime has cost £11bn since the Conservatives took office in 2010.

“The Tories are creating a perfect storm of failing communities that don’t feel safe, failing to tackle violent crime and failing to provide the ambition and opportunities young people need to make positive choices,” he said.

Dick’s departure followed a barrage of criticism of the force including over its handling of the case of Sarah Everard, who was raped and murdered by a serving Met officer, and recent revelations of officers bragging about violence towards women and exchanging racist and Islamophobic messages.

Patel was angered by the London mayor’s failure to inform her that he had called Dick to a meeting on Thursday afternoon, which she considered “unprofessional”.

The Northern Ireland secretary, Brandon Lewis, claimed on Sunday that Khan had played politics to force Dick out. He told Times Radio: “I think he should’ve been consulting with the home secretary, bearing in mind this is a man who just a couple of months ago extended Cressida Dick’s contract.”

Asked if he thought the mayor was playing politics, he responded: “Possibly, to be frank.”

Responding to Lewis’s comments, a source close to Khan said it was the mayor’s job to hold the Met to account. They said: “It is absolutely right that he ensures it has the right leadership to command the trust and confidence of Londoners.”

Pressure is building on Patel and Khan to agree on who will replace Dick, amid reports that the home secretary could look abroad for a successor. The Mail on Sunday claimed that among the shortlisted officers were the Queensland police commissioner, Katarina Carroll.

By convention, the commissioner is drawn from the ranks of serving senior officers in the UK. Home Office sources said they could recruit from abroad but it was likely the next commissioner would come from the UK.

In the Observer this weekend, Khan wrote that he would not support the appointment of a new commissioner “unless they can clearly demonstrate that they understand the scale of the cultural problems within the Met and the urgency with which they must be addressed”.

The mayor’s office for policing and crime is in discussions with the Home Office about the process to recruit the next commissioner. Sources close to the mayor indicated that he remained “open-minded” as to who that person might be.

Whoever follows Dick will have to deal with “a real cultural issue” in the force that cannot be fixed overnight, according to the former HM inspector of constabulary Zoë Billingham. She told Sky News’s Trevor Phillips on Sunday: “Whoever the incoming commissioner is, they will have a very very full inbox. They will have precisely the same problems that Cressida was contending with to deal with.”