Theresa May’s opening bid of putting security on the Brexit negotiating table will be viewed by defence, intelligence and police chiefs across Europe – and even within the UK – as both surprising and brutal.

The president of the European parliament, Antonio Tajani, able to speak more openly than officials, responded relatively politely, saying “close cooperation on defence, police, intelligence and action against terrorism” should continue whether there is a deal or not. The European parliament’s Brexit coordinator, Guy Verhofstadt, was less restrained, throwing in the word “blackmail” and saying security should not be used as a bargaining chip.

At the heart of the threat is the pre-eminence in Europe of UK intelligence-gathering on terrorism, crime and perceived international threats such as Russia. Much of this intelligence is routinely shared with partners elsewhere in Europe.

One of the biggest problems for May is that the threat will be seen not only as reckless – using European-wide sharing of intelligence on terrorism for leverage in a trade deal – but that it lacks credibility. The former permanent secretary to the Treasury, Sir Nicholas Macpherson, said as much: “Not a credible threat to link cooperation to a trade deal.”

Intelligence agencies across Europe are going to continue sharing information about terrorist plots whatever Brexit deal emerges. The UK police will continue to work with counterparts elsewhere in Europe. And European defence is not a European Union issue but one for Nato, the US-led alliance.

It is almost inconceivable that any of the UK intelligence agencies – MI5, MI6 or GCHQ – picking up any hint of a terrorist attack being planned on the continent would hesitate to pass it on.

But that works both ways. France, Germany or any other European intelligence agency might not have had anything to pass on about last week’s Westminster terror attack – no evidence has emerged yet that Khalid Masood had any links to European jihadis – but they have informed the UK often in the past about visits or communications involving either British extremists on the continent or European jihadis in the UK.

Although Downing Street later tried to play down May’s words, insisting there was no implied threat, the prime minister’s letter is explicit. “In security terms, a failure to reach agreement would mean our cooperation in the fight against crime and terrorism would be weakened,” she wrote.

The home secretary, Amber Rudd, supposedly sent out to dampen down the issue in an interview on Sky, simply added to the sense that security was now a negotiating tactic. She said, rightly, that the UK was the biggest contributor to the European law enforcement agency, Europol, in terms of supply of information but added an explicit threat: “If we left, we’d take our information with us,” she said.

Raffaello Pantucci, the director of international security studies at the London thinktank the Royal United Services Institute, said there could be a benign interpretation: that May was just flagging up that the UK is a big actor in terms of European security. He estimated that about 40% of the information going to Europol came from the UK and the European counter-terrorism strategy had been lifted almost word by word from the UK’s one.

But he added: “I like to hope and think this is not a direct threat that the UK would turn a blind eye to incidents on the continent. It is a bit surprising.”

The UK is likely to pull out of Europol, which is at present headed by a Briton, Rob Wainwright, a former MI5 analyst. Giving evidence to the Commons home affairs select committee this month, he said that European crime-fighting would be weakened if the UK was to leave. But the UK would also be a loser, he said, citing the European arrest warrant scheme, which he said had allowed the UK to send 2,000 criminals back to other European countries since 2004.

“That’s 2,000 criminals fast-tracked out of Britain, and if we lost that capability it is unlikely to be nearly as effective,” Wainwright said.

The UK has embraced, alongside the US, the idea that Russia has emerged as a major security threat, particularly to the Baltic states. Just this month, the UK deployed troops to Estonia for the first time since the end of the cold war. But that is as part of Nato. There is no European defence force. No one in the UK Ministry of Defence, where Nato is regarded as the cornerstone of UK defence, would contemplate for a nanosecond doing anything that would undermine Nato.

The head of the UK’s overseas spy agency, MI6, Alex Younger, in a rare speech in December, addressed the issue of Brexit. The threats from terrorism and elsewhere existed before Brexit, he said, and the joint capabilities to defeat it had existed before it too.

“The need for the deepest cooperation can only grow. And I am determined that MI6 remains a ready and highly effective partner, just as the UK is and will be. These partnerships save lives in all of our countries,” Younger said.