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New cars are seen in a parking lot at the Jaguar Land Rover plant in Liverpool, northern England.

Phil Noble/Reuters

The wrong Brexit deal could cost tens of thousands of car jobs and risks production at Britain’s biggest car maker, Jaguar Land Rover, its boss warned, adding he had no idea if his plants could function after Britain quits the European Union.

Ralf Speth said the company would not be able to build cars if post-Brexit customs checks meant that the motorway to and from the southern English port of Dover, which is used to transport components, becomes a “car park” because of snarl-ups.

Mr. Speth was speaking on Tuesday at a conference shortly before Prime Minister Theresa May, who is battling to have her “Chequers Brexit” plan accepted by many in her Conservative Party as well as the EU itself before Britain’s exit on March 29.

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“A thousand [jobs were] lost as a result of diesel policy and those numbers will be counted in the tens of thousands if we do not get the right Brexit deal,” Mr. Speth said, referring to redundancies made earlier this year at the firm.

“Currently I do not even know if any of our manufacturing facilities in the U.K. will be able to function on the 30th [of March],” he said.

London and Brussels say they want to get a divorce deal at the Oct. 18 EU Council meeting or at the latest by the end of the year.

Under Ms. May’s proposals, Britain will seek a free trade area for goods with the EU, largely by accepting a “common rulebook” for goods and British participation in EU agencies that provide authorizations for goods.

Asked about Mr. Speth’s comments, Ms. May’s spokesman said her plans include specific proposals to protect jobs in industries using so-called “just-in-time” supply chains.

But that plan has angered hardline Brexiters in her party.

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Worker George Baker looks at a build card for a vehicle destined for China at the Jaguar Land Rover facility in Solihull, Britain, Jan. 30, 2017.

Darren Staples/Reuters

Britain’s car industry, which employs more than 850,000 people, is overwhelmingly foreign-owned and dependent on supply and distribution chains which spread around the world, but especially the EU, its largest export market.

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Mr. Speth said unfettered access to the single market for Jaguar Land Rover (JLR), which built nearly a third of Britain’s 1.67 million cars last year, was “as important a part to our business as wheels are to our cars.”

He also warned that long-standing issues around low productivity in Britain could be compounded by a Brexit agreement which made the country less competitive.

JLR, which is due to open a new plant in Slovakia later this year, said it was already cheaper to build abroad.

“It is thousands of pounds cheaper to produce vehicles for instance in eastern Europe than in Solihull [in central England], and what decisions will I be forced to make if Brexit means not merely that costs go up but that we cannot physically build cars on time and on budget in the U.K.?” he said.

“Any friction at our borders puts our production in jeopardy – at a cost of £60-million a day."