Thousands of international lorry drivers are braced to spend Christmas Day cooped up in their cabs near Dover as it was announced that hundreds of soldiers would join the repatriation effort to get them home.

It also emerged that a team of French firefighters have been sent to Dover to help clear huge queues of lorries waiting to cross the Channel after the British government failed to provide sufficient coronavirus testing capacity.

Ministers had promised to get the queues moving by Wednesday morning after drivers were left waiting in their cabs for days. But it has emerged that fewer than 100 of the thousands of waiting vehicles were able to cross the Channel that day.

The government said more than 2,000 tests had been carried out by Thursday lunchtime, but the Port of Dover said only 700 lorries had managed to begin the crossing.

While it had been clear by Tuesday that French authorities would require a negative test result from each of the thousands of drivers waiting to cross, the port said the UK testing effort initially operated only at Manston airfield, one of the sites where queues formed, and that problems getting the testing kits there had severely restricted capacity.

It took until Thursday to get the tests out to the port and the surrounding roads where many more of the lorries were waiting, it said.

The transport secretary, Grant Shapps, said 2,367 tests had been carried out as of midday on Thursday, with 2,364 negative results and three positive. On Thursday night, he said: “We need to get the situation in Kent, caused by the French government’s sudden imposition of Covid restrictions, resolved as soon as possible. I have today sent special instructions to the Army to take control of testing and HGV management operations in the county. Our aim is to get foreign hauliers home with their families as quickly as we can. I know it’s been hard for many drivers cooped up in their cabs at this precious time of year but I assure them that we are doing our utmost to get them home.”

Shapps earlier said the border at Dover, the Eurotunnel and Calais would remain open throughout Christmas to get hauliers cleared “as soon as possible”, with the help of 10,000 more tests brought in by 26 French firefighters on Thursday morning.

Shapps promised that ferries would sail on Christmas Day and Boxing Day to help unblock the logjam caused by a 48-hour border closure imposed by France in an attempt to stop the new hyper-infective Covid variant from crossing the Channel.

France’s ambassador to the UK, Catherine Colonna, tweeted that the two countries were “neighbours, partners, allies and (yes) friends”. She posted a picture with the hashtags #StrongerTogether and #thursdayvibes showing the orange-jacketed firefighter, or pompiers, carrying out Covid tests on hauliers before dawn on Thursday.

Some drivers have already spent nearly a week stranded due to the diplomatic impasse. More than 300 soldiers will be brought in to take charge of testing and lorry marshalling to clear the backlog.

The government said catering vans would be brought in to provide complimentary hot food and drinks to stranded hauliers at Manston, with Kent council and volunteer groups providing refreshments to those stuck on the M20.

The Road Haulage Association (RHA) estimated that between 8,000 and 10,000 lorries had been caught up in the queues that started forming around ports on England’s south coast when the border was closed on Sunday. The government said it believed the number to be closer to 6,000, though ministers have previously vastly underestimated it.

A Port of Dover spokesman confirmed ferry services would continue running throughout the night and on Christmas Day to help ease congestion. Traffic is moving more quickly at the Eurotunnel, where more than 1,000 vehicles left on Wednesday night, with around 2,000 more expected to depart by the end of Thursday. But many will remain there for Christmas Day, according to Duncan Buchanan, a policy director at the RHA. He said: “The most reassuring thing is that food is getting through at Manston, and I have to say a big thank you to everyone who volunteered to help drivers stick it out in cold conditions in the days leading up to Christmas.”

The Department for Transport said French authorities had been called in not because of failures on the UK’s part, saying they were there as part of a collaborative effort. The French embassy in the UK was contacted for comment.