Labour’s deputy leader has said Britain is on the frontier of a “new cold war” that Russia is winning and discussed a claim that Theresa May may have blocked an intelligence services investigation into the Russian dealings of Arron Banks, the businessman behind the Leave.EU campaign.

Tom Watson made the remarks at an Observer event at the Labour conference in Liverpool on Sunday where he accused the government of being wilfully blind to Russia’s attack on British democracy.

He said Britain had to wake up to the scale of the Russian threat, which he called a “Blitzkrieg for the digital age”. He also said it was imperative that Britain launched an inquiry into what happened in the referendum similar to US special counsel Robert Mueller’s probe into whether the Trump presidential campaign had links to Moscow.

Watson singled out May for criticism and demanded that she respond to allegations he had heard that she had thwarted a security services investigation into Banks’s dealings with the Russian government, which were revealed by the Observer two months ago.

Banks has consistently denied receiving money from Russia, but the source of his wealth has been under scrutiny since he gave £9m to Leave.EU, the largest political donation in British history.

Revealed: Leave.EU campaign met Russian officials as many as 11 times Read more

“Did [May] ask the security services to investigate? Or did she stop them doing so?” Watson said. “There is a suggestion that in the run-up to the referendum the prime minister – in her capacity at the time as home secretary – declined at least one application from the security services to mount a full investigation into Mr Banks and others suspected of Russian influence. We need to know if that is true.”

When asked about his remarks later, he said: “The intelligence services reported to her. It would be highly irregular if they hadn’t reported on Banks’s trips to the Russian embassy.”

He failed to say where he had heard the suggestion that May had refused permission for Banks to be investigated.

The Electoral Commission is currently investigating the source of the multimillion-pound donations and loans Banks gave the Leave.EU campaign but Watson said that it was vital that a body with the powers to subpoena witnesses and evidence now took it over: “He says the money came from ‘his bank account’. But who put it there?”

Watson said the shadow digital minister, Liam Byrne, and Darren Jones, MP for Bristol North, were launching a parliamentary petition to get the subject of a Mueller-style inquiry into the referendum debated in the House of Commons and that he had already raised his concerns with Labour’s ruling body. “I really hope that we as a party understand what is at stake,” he said.

Facebook Twitter Pinterest Whistleblowers Shahmir Sanni and Christopher Wylie at the Observer event. Photograph: Christopher Thomond/The Guardian

Watson said May talked “tough” on Russia but claimed it was a different story behind the scenes. “All of us need to wake up to the true scale of what’s been going on,” he said. “Now is the time when the government must drop the three wise monkeys act – see no evil, hear no evil, speak no evil – about the biggest threat to the integrity of our democracy.”

He accused the government of failing to adequately respond to the findings of the select committee for digital, culture, media and sport’s interim report into fake news, which examined Russia’s role in spreading disinformation and influencing political campaigns.

Damian Collins, the Conservative MP who is chair of the committee, said it was essential that the government reveal whether the National Crime Agency was investigating Banks’s finances and dealings with the Russian government in relation to Leave.EU and the Brexit referendum.

The panel also featured Christopher Wylie, the Cambridge Analytica whistleblower, and Shahmir Sanni, who revealed how Vote Leave broke the law by massively overspending. Wylie described how he was helping the FBI with its investigation into Cambridge Analytica and said he’d been asked far more questions about Brexit by US authorities than by British ones.

No 10 has been contacted for comment.