They have prepared for potential riots, looting and disorder if there are long queues at ports or shortages to food, petrol and medicine.
Senior officers at the National Police Coordination Centre said the delay created fresh “complexity” in their operations, as well as an increasing risk of angry protests.
While there is no specific intelligence of potential violence, police leaders said they were prepared for “worst-case scenarios”.
More than 10,000 officers trained in public order are available to be rolled out in a 24-hour period needed – more than the total used in the 2011 riots – with the first wave of 1,000 able to mobilise in an hour.
The new National Police Chiefs’ Council (NPCC) chair, Metropolitan Police assistant commissioner Martin Hewitt, said the situation would depend on “the way prominent people with a platform conduct themselves”.
He added: “We’re in an incredibly febrile atmosphere as a result of the whole EU exit scenario. All groups of people need to think carefully – we are in a febrile atmosphere and if you’re in a position where you know you’re going to be listened to, you need to be very careful about the language you are using so it doesn’t end up with consequences that weren’t intended.”
Mr Hewitt said his warning applied to both politicians and activists, in the wake of rhetoric accusing MPs of “betraying Brexit” and protests that saw Brexiteers drag effigies of politicians through the streets.
MPs have been called “traitors” by groups of “yellow vest” protesters stationed for months outside parliament, and police have been contacting politicians to ensure they have “appropriate, proportionate and effective security advice and security arrangements” in place.
Commander Adrian Usher, who leads on MP security at the Metropolitan Police, said more MPs had been requesting “security packages” over safety fears.
“We have seen a greater level of abuse reported,” he added. “As we’ve moved towards key dates in the Brexit calendar there have been spikes ... elected representatives must be able to express themselves and go about their business without facing abuse or harassment.”
In the past two weeks, police have recorded 37 crimes directly related to Brexit across England and Wales, including malicious communications, verbal abuse, harassment and offences committed at protests.
Officers also predict another rise in hate crime, following a large spike in the wake of the 2016 EU referendum.
Chief Constable Charlie Hall, the NPCC lead for operations, said: “We know there are strong and competing views and a really loud debate but underneath that, there remains absolutely no excuse for hate crime and EU exit should not be used as one.”
He said police were facing high demand from violent crime and other issues, and had asked ports, border authorities and companies involved in food, petrol and medicine supplies to make their own contingency plans for no deal.
“Policing support should only be called on if absolutely necessary,” he added. “We have met with the Ministry of Defence to discuss the potential roles it could play to provide support for policing.”
At least 15 regional forces put restrictions on leave in preparation for Brexit, and others may have to extend shifts, and stop training and peripheral policing work if their officers need to be deployed elsewhere.
Two forces have officially asked for mutual aid – Kent, which covers the Port of Dover, and Hampshire, which covers Portsmouth and Southampton.
Mr Hall told The Independent preparations had so far had little effect on day-to-day crimefighting, but if national mobilisation plans are activated “the impacts could be more significant”.
Around 1,000 officers have been trained in tactics only used in Northern Ireland, including the use of armoured vehicles and water cannons, in case officials in Belfast request extra support from England and Wales.
The extra cost is expected to run into the millions of pounds, but the total will not be known until police chiefs bill the Home Office.
As stalemate continues in Westminster, Mr Hewitt said police still “do not know what that exit will look like” and still face losing access to EU databases and law enforcement tools overnight.
Mr Hall said officers were “watching the politics closely” after their plans for what was originally to be Brexit day were stood down.
“We stood down some resourcing and continue to review what forces need,” he added. “Our plans are sufficiently flexible to adjust to whatever dates are thrown at us," he added.