The security minister admitted this week that a deal has not been struck with the EU and said negotiations were at a “sensitive point”.
Responding to concerns about information sharing on potential terrorists, James Brokenshire told MPs: “If negotiations do not conclude successfully, we will move back to pre-existing tools and powers.”
Ch Supt Griffiths said police would put themselves in the “best position possible”, but warned of “challenges and unintended consequences” of any change.
“In theory, we should be able to exchange information with EU partners fairly quickly because we’ve all got a vested interest in the safety of citizens,” he added.
“But how to do that is the challenge – the efficiency and effectiveness of the systems.
“I’m not sure we fully understand the consequences in terms of information transfer from other countries in Europe.”
Ch Supt Griffiths said that crime, particularly the most serious forms of organised criminality, was becoming more “trans-national”.
Police scrambling to save suspects’ details for fear of no-deal Brexit
He suggested that the current model of having 43 operationally independent regional forces in England and Wales should be reformed.
“Some of serious and organised crime is global and we can’t escape that,” he added.
“The boundaries we have adhered to for decades are fruitless now relating to some of the criminality going on that can originate abroad and hit your local neighbourhood.”
But the main database used to access intelligence on convicted extremists and other criminals, the Schengen Information System (SIS II) is among those that the UK faces losing access to.
British police officers checked it 539 million times in 2017 alone, and their equipment currently searches SIS II and the Police National Computer simultaneously.
Britain will also be cut out of the European Arrest Warrant system, which allows wanted suspects to be extradited to Britain, and for those on UK soil to be arrested and ejected from the country.
The law would have to be changed to allow the alternative Interpol red notices to be used to detain suspects without going to court for a warrant.
Also at stake is Britain’s membership of Europol, a pan-EU policing body used to conduct joint operations against international extremist and organised crime groups, and pool intelligence.
Access to the European Criminal Records Information System (ECRIS) could be axed and agreements on the sharing of passenger name records, and DNA, fingerprint and vehicle registration data must also be reached.
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Last month, cabinet minister Michael Gove suggested that the government would not accept a deal where the UK had to “accept the jurisdiction of the European Court of Justice”, even if it meant losing SIS II and other tools.
As Ms May, the former prime minister and home secretary, mouthed “what” and shook her head, he claimed: “When it comes to everything – security and other matters -no deal is better than a bad deal.”
The former head of MI5 previously said there was no “security upside” to Brexit and the best the government can hope to do is minimise its negative impact.
Lord Jonathan Evans warned that although intelligence sharing would be largely unaffected by Brexit because of long-standing bilateral relationships, that was “only a small part of the overall picture”.
The National Police Chiefs’ Council announced that preparations for the potential loss of EU tools and powers were complete last October.
At the time, its lead for Brexit warned that the contingencies developed were not “like-for-like replacements”.
Deputy Assistant Commissioner Richard Martin added: “In all cases the replacements are slower, less effective, and more bureaucratic for officers than our existing setup.
“Existing EU tools allow us to respond quickly and intelligently to crime and terrorism impacting the UK and the EU – they make us better at protecting the public. We want to avoid leaving without a deal because that would see us lose access to those important tools.”
Police have also planned responses to worst-case scenarios including mass protests, disorder and chaos at ports, and many forces have banned annual leave over the Brexit period.