Theresa May will be seen as “soft on terrorism and organised crime” unless she signs the UK up to continued membership of Europol, the EU’s law enforcement agency, warns the former deputy prime minister Nick Clegg.

A key decision over whether the UK remains inside Europol, which pools criminal intelligence and allows this to be shared between police and security forces across the EU, is seen by MPs as the crucial “first test” for the prime minister in Brexit negotiations.

MPs of all parties – including those on the cross-party European scrutiny select committee – are demanding a full House of Commons debate and vote on Europol within the next month as a matter of urgency, following calls from police and intelligence agencies for the UK to retain full membership.

May herself has praised Europol and other EU crime-fighting agencies for bringing terrorists to justice and helping to track down cross-border criminal networks, including those involving paedophiles and people traffickers.

Clegg said last night that the “hugely important” decision would show whether May was ready to put the national interest and safety of UK citizens above the “anti-EU” dogma of some in her own party who wish to leave all organisations run by the EU, even if they clearly benefit the UK.

“If the government elevates anti-European dogma above what has been proven to keep Britain safe, Theresa May will leave herself open to claims that she has gone soft on the fight against international terrorism and soft on the fight against organised crime,” Clegg said.

“It would be a spectacular own goal if we were to leave an organisation that we helped establish, contribute so much towards, and which has proved so vital in the fight against international criminal networks.”

Sir Keir Starmer, Labour’s Brexit spokesman, said it was vital that cooperation with EU nations was maintained in the fight against terrorism and international crime: “That will be difficult now that we are exiting the EU. I would expect the government to set up a specialist team to negotiate this aspect of Brexit, and parliament should be kept fully informed at all times. These issues require ongoing cross-party support.

“Opting into the latest Europol measure is the first test of the government’s resolve on this issue. The clock is ticking but so far the government has sat on its hands. This is not an issue that can be fudged to appease Tory backbenchers.”

Ministers must decide by the end of this year whether to accept a new, expanded remit for Europol, agreed by the 28 member states after the Paris and Brussels terror attacks. If they do, the UK police and intelligence services will continue to have full access to Europol databases and services. However, many Tory MPs will see the move as a “fudge” and a sign that the prime minister is backing away from a “hard Brexit”. Crucially, if the UK stays in Europol it will have to pay budget contributions to the EU and accept a role for the European Court of Justice in resolving disputes – two conditions unacceptable for hardcore Brexiters.

Were the government to reject the deal, the UK would drop out of Europol from May next year. The many UK staff at its headquarters in The Hague, including its British director, Rob Wainwright, would have to go.

May has raised the issue with international leaders in recent weeks, expressing her preference for remaining in Europol but also her fears that her party will see such a move as an early Brexit capitulation. In a speech in April, before the Brexit vote, she cited crime-fighting and sharing intelligence at EU level as a major reason for staying in the EU.

“Our response to Paris and Brussels cannot be to say that we should have less co-operation with countries that are not only our allies but our nearest neighbours,” she said.

In a recent report, the Commons European scrutiny committee demanded a debate in parliament before any decision is made and expressed disappointment that ministers had said so little on the subject.

The Home Office said that no decision had yet been taken: “The prime minister has stated that law enforcement cooperation will continue when the UK is outside the EU, and we will do what is necessary to keep our people safe. The Government will take a decision on whether to opt-in to the new Europol Regulation in due course.”