Military planners have been deployed to the Department for Transport, the Home Office and the Foreign Office as officials desperately try to avoid backlogs and chaos at the border in the event of a no-deal Brexit, the Observer can disclose.

Details released under the Freedom of Information Act reveal that 14 military planners have been dispatched by the Ministry of Defence to key ministries, which also include the Cabinet Office, the hub of the government’s Brexit planning, in a sign of concerns inside Whitehall at the prospect of Britain crashing out of the EU with no agreement in place.

In a ramping up of no-deal preparations in recent weeks, one planner is in place at Chris Grayling’s beleaguered DfT, which has already been criticised for awarding a £14m contract to run ferries in the event of a no-deal Brexit to a company that does not have any ships.

The transport secretary has also been criticised for a live rehearsal of emergency traffic measures that will be put in place to prevent logjams around Dover in such a scenario.

Some drivers taking part in the event described it as a waste of time, while haulage campaigners said it was “too little, too late”.

Four planners have been posted in the Border Force, which is facing the challenge of keeping passengers and goods flowing to and from Britain should no EU agreement be signed. Three are operating in the Foreign Office, while six are working from the Cabinet Office.

The departments involved refused to comment on why they had requested a military planner, or what projects they were assisting. A Defence ministry spokesman said: “The MoD routinely works with other government departments on planning for a range of contingency scenarios.”

Insiders said some departments had asked for assistance on no-deal planning, “recognising the unique skills and operational planning experience the military can offer”.

Exercise planning and overall “command and control” advice are understood to be their main duties.

Cabinet ministers were in open disagreement last week over the consequences of leaving the bloc with no deal in place, with business secretary Greg Clark saying such an outcome would be disastrous, but defence secretary Gavin Williamson insisting that Britain could succeed under any Brexit scenario.

Contingency no-deal preparation has been taking place under the codename Operation Yellowhammer, with plans being drawn up on the assumption that critical trade between Calais and Dover will face disruption.

It has been suggested that Michael Gove, the environment secretary, was preparing to request the help of a military planner to help ensure food supplies were not disrupted. However, details passed to the Observer suggest he has not yet done so.

Speaking in the Commons last week, Liberal Democrat leader Vince Cable said he had been told by civil servants of contingency plans to slaughter a third of all British sheep in the event of no deal.

World Trade Organisation tariffs would hit exports so hard that there would be too many sheep, causing domestic prices to plummet, he said.

“We have the problem that if we leave the European Union with no deal, on WTO terms, the European Union’s tariffs on dairy products, lamb and various other items, which are quite high, immediately kick in,” Cable said.

“The problem with that, as we discovered when we had the foot and mouth epidemic, is that if we cannot export, prices crash. The only logical response from the farming industry, in order to maintain the value of the stock, is to slaughter large herds. This will happen.

“We know there is a paper at the moment in the agriculture department – the Department for Environment, Food and Rural Affairs – setting out a plan for slaughtering a third of all British sheep in order to maintain the integrity of the market. That is an inevitable consequence of a high tariff obstructing British exports.”

But a Defra spokesman played down the suggestions that a mass slaughter was on the cards in the event of a no-deal.

“As the environment secretary has made very clear, a widespread cull of sheep, as has been suggested by some, is not something that government anticipates or is planning for.”

Government sources said “significant work” was under way to ensure UK exporters could maintain access to EU markets after 29 March. “As part of this we are in close contact with the sheep sector,” one said.

Fraught no-deal planning is also taking place in the corporate world, with wildly different problems facing different sectors. Brexit consultants said the food and textile sectors would face the highest increase in potential customs duty. Several companies are said to be looking at potentially paying tens of millions extra in duties and administrative costs each year.

Meanwhile industry sources said supermarkets are building up a couple of months’ stock, locking up several hundred million pounds. Big pharmaceutical companies are generally spending in the range of £5m to £20m on preparation, with banks spending even more.

Some companies are being warned that they are stockpiling too much and that they may be better off ensuring they have cash in the bank to deal with an unpredictable market.

James Stewart, head of Brexit at the accountancy firm KPMG, said: “Confidence is thinning and we continue to see heavy client activity on no-deal planning. We think everyone should be looking at contracts, supply chain security and workforce planning. Even at this stage we are still seeing massive variations in preparedness.

“Companies who sell in sterling and buy in dollars could be in real trouble if they haven’t hedged correctly. A raft of businesses in automotive, retail and hospitality remain very exposed.”