The UK’s full participation in European Union security and intelligence co-operation will be critical to the fight against terrorism after Brexit, leading British security experts have said, as Theresa May announced wide-ranging new plans to counter extremism.

The growing demands for the prime minister to face down anti-EU forces in the Tory party and make membership of bodies such as Europol, the EU’s criminal intelligence agency, a top priority, came amid fears that Brexit could leave the UK with inferior access to key European databases and deprive British police forces of vital tools in high-level, pan-European anti-terror probes.

The calls – from senior figures including Sir Hugh Orde, former chief constable of the police service of Northern Ireland and former head of Europol Max-Peter Ratzel – were reinforced on Saturday night by Dominic Grieve, the Tory chair of the Commons intelligence and security committee.

Grieve said full participation, even if it meant accepting EU rules and judicial oversight for the European Court of Justice (ECJ), could not be more crucial. He said he believed May was committed to remaining closely involved in EU security but believed that doing so would require compromise that would be hard for some in May’s party to accept.

“Although our partnership with the US for intelligence sharing is extremely important, the fact is that the current terrorist threat is very much a European dimension issue. The Schengen database and knowing about who has moved where are all intimately dependent on European systems and we have got to try to remain in them,” he said.

“I think it is going to be very difficult, and the government and Parliament will have to face up to the fact that it might turn out during the Brexit negotiation that some sort of mechanism for the European court is going to be necessary,” he added.

After the Manchester attack, which killed 22 people and left dozens of others grievously injured, it was revealed that suicide bomber Salman Abedi had travelled back to England from Libya via Turkey and Dusseldorf four days before the attack.

The bomb contained the same kind of explosives as those used in the Paris and Brussels attacks in 2015 and 2016, prompting speculation that Abedi was linked to terror cells on the European mainland. UK counter-terrorism officials investigating the Abedi network are working with Europol, with separate bilateral inquiries ongoing with German and Belgian intelligence agencies.

Sir Hugh Orde says Britain must remain a member of EU bodies. Photograph: Miguel Medina/AFP/Getty Images

Orde told the Observer that UK membership of EU bodies such as Europol and Eurojust, which brokers judicial co-operation in criminal cases, not only allowed access to huge amounts of vital data, but also meant UK police could set up joint inquiries with German police or those from other national forces without delay.

European arrest warrants were also essential, he said: “If we don’t have all this, it makes it a lot more difficult to do this crucial work. It it vital that we get to a situation as close to what we have as members of the EU as possible, though it is difficult to see how we do that.” As members of Europol and Eurojust, states are subject to EU rules on data transfers and procedures and ECJ jurisdiction in any disputes or appeals over use of tools such as arrest warrants.

Ratzel, who led the pan-EU crime-fighting agency until 2009, warned that Britain must accept some of the rulings of the ECJ as part of any deal – something May has previously ruled out. “My expectation is that once you have an important role to play, you have to be under the European Court of Justice. Cherry picking cannot work. You have to obey the rules of the club. I am personally sure that the Europeans will never accept Britain having a role in Europol without being under the rule of the European Court of Justice.”

As anti-terror policies of the rival parties were placed centre stage in the election campaign, May declared on Saturday night that more needed to be done.

She added: “We need to be stronger and more resolute in standing up to these people.” Spelling out details of a new commission for countering extremism, announced in the Tory manifesto, she said government had to widen its role and approach.

“There is clearly a role for government in tackling extremism where it involves behaviour that is or ought to be criminal. But there is also a role for government to help people and build up organisations in society to promote and defend Britain’s pluralistic values, and stand up to the extremists.”

May’s initiative came as Labour also attempts to boost its security credentials by pledging to appoint 1,000 more intelligence staff. Jeremy Corbyn has already said he would appoint an extra 10,000 police, 3,000 firefighters and 3,000 prison officers. Suggesting that public spending cuts had put people in danger, Corbyn said that ensuring the safety of communities “demands properly resourced action across many fronts.

“It means upholding and enforcing our individual rights, promoting community relations, supporting our emergency services, tackling and preventing crime and protecting us from danger, including threats of terror and violence,” he said.

The Manchester bombing was the worst terror attack to hit Britain since the 7 July attacks in London in 2005.

A street in Manchester’s Moss Side was evacuated by counter-terrorism officers and controlled explosives were used in a raid on a property in Cheetham Hill.

Two men aged 20 and 22 were arrested in the early hours of Saturday morning at the Cheetham Hill address, police said. Neighbours identified Yahya and Mohamed Werfalli as two of the occupants of the raided house. They were said to be of Libyan descent and part of the same friendship groups as the Manchester bomber.