It could take days to clear the backlog of lorries waiting to cross to France, Transport Secretary Grant Shapps says.
Drivers have been warned they may have to spend Christmas in their cabs.
About 4,000 lorries are parked at Manston Airport in Kent, with another 2,300 being held on the M20 after the French closed their border with the UK.
Crossings will continue over Christmas but all drivers are required to test negative for Covid-19 before being allowed into France.
Tests are being administered by members of the military and French firefighters.
Polish defence minister Mariusz Blaszczak said in a tweet that a team of territorial soldiers was being sent to the UK to support the testing effort.
Of the 2,367 drivers tested by 12:00 GMT, three have tested positive, Mr Shapps said.
"Spending days in a lorry on your own puts you in an extremely low risk category," he added.
At Manston for Christmas
Croatian driver Ante Kostelac told the BBC: "All hopes for Christmas with the family are over.
"I'm still waiting for testing at the airport, when asked when it will be, I got the answer 'who knows'."
"I'll obviously be at this airport for Christmas."
Operation Stack has been put in place on the M20 allowing lorries queuing for the Port of Dover and Eurotunnel terminal to be held on the motorway.
The Port of Dover said fewer than 100 freight vehicles had passed through overnight due to the restrictions on testing.
"However, now testing has fully mobilised at the port we anticipate that figure rising significantly throughout the day," a spokesman said.
As well as ensuring ferries will now sail on Christmas & Boxing Day, we've also got great cooperation by French firemen working with NHS Test & Trace and our brilliant military in a big effort to clear the backlog created by the French border closure: https://t.co/5Q22QyI58k— Rt Hon Grant Shapps MP (@grantshapps) December 24, 2020
Mr Shapps said: "It will take several days to clear the backlog."
"The one thing that people can do is not turn up in Kent; it won't get you through faster, indeed it will make things more difficult on the ground.
"The testing is happening and the negative tests are coming through now, and with police's help clearing the entrance to Dover, the traffic is able to move, and with Eurotunnel's help, things are just starting to move.
"It's never going to be a quick operation."
The head of the Road Haulage Association said he sympathised with the small number of lorry drivers who clashed with police in Dover on Wednesday.
Richard Burnett said: "I really feel for these drivers that have ended up being pawns in a larger game.
"Are they going to be held here until Boxing Day or beyond?"
At the scene: Amanda Akass, BBC South East
Testing is under way at the Port of Dover with traffic slowly moving through this morning.
Many of the drivers have spent two, three or even four nights in their cabs with no toilet or washing facilities.
It's not just hauliers either. There are hundreds of vans full of workers from across Europe and families with small children trying to get home in time for Christmas.
Some drivers queuing through the town centre told me they've only moved 100m since the port reopened. I saw a few arguments breaking out at junctions where drivers were perceived to be pushing in.
The main routes through Dover are still largely blocked although there are definitely more police around today redirecting traffic away from the gridlock. Local volunteers are also out in force delivering much-needed food and drink to the waiting travellers.
As part of Operation Stack, the coast-bound carriageway of the M20 is closed between junction eight and the A20 at Hawkinge.
The London-bound carriageway is also shut between junction nine for Ashford and junction eight for Leeds Castle, with traffic being diverted on to the A20.
All EU freight traffic heading to the Port of Dover and Eurotunnel Terminals is being redirected to Manston Airfield, Highways England said.
Freight traffic on the M25 is being told to use the M2 and A2.