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Thousands of illegal immigrants will enter Britain from France after a no-deal Brexit, ministers have been warned.

A Whitehall analysis concluded that desperate migrants will inevitably slip on board food and goods lorries held up at Calais by new checks when Britain leaves the smooth-flowing European Union trade area.

The alert was revealed as the Cabinet met this morning to ramp up no-deal preparations, including the appointment of hundreds of extra officials to protect Britain’s borders and attempt to speed up traffic flows.

With just 101 days left until the official date for Brexit on March 29, ministers were expected to press the button on an 11th-hour spending spree to counter the risk of goods and medicines running out amid cross-Channel gridlock.

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Among emergency measures, the military may be tasked to take over some police duties to free up officers to keep order in areas facing traffic chaos.

The Health department is chartering planes to ferry medicines, while the Environment department is urging food companies to stockpile ingredients that could run short.

The Transport department has the biggest headache, following warnings that the area around Dover will face chaos as lorries queue for new regulatory checks.

Farmers’ leaders today warned that a no-deal Brexit would be “disastrous” and threaten to hit supplies of seed, animal medicines and workers.

The alert about illegal migrants was issued by Michael Gove’s Department for the Environment which is responsible for food imports, most of which are carried in lorries that are already targeted by people smugglers and would-be asylum seekers.

Officials warned that hold-ups on the French side could lead to an influx of stowaways. Former immigration minister Damian Green told the Standard that stowaways were a regular problem. “If there are problems at Channel ports that delay lorries in France, its easier for asylum seekers to climb on board and get to Britain illegally.

“This is another reason why no-deal poses a threat, not just to our economy but to our borders as well. It’s absolutely essential that we maintain the goodwill of the French authorities in stopping refugee camps building up on the other side of the Channel.”

The National Farmers’ Union appealed to farmers to write to their MPs — many of them hardline Brexiteers — and urge them not to back the UK crashing out of the EU in March. Amid growing anger at the political chaos in Westminster, NFU president Minette Batters warned that no-deal would be “catastrophic for farming”.

“The more we have seen, the more disastrous it looks,” said Ms Batters, who runs a family farm in Wiltshire. “Ninety-five per cent of our animal medicines and vaccines are currently produced in the EU, many of our seeds are coming from the EU, our fertilisers, water chemicals, vital things that are coming from the EU at the moment.”

Meanwhile, more than 50 business figures have signed a letter calling on Theresa May to back a second referendum to break the deadlock if she cannot persuade MPs to back her Brexit deal when the vote takes place next month.