Sir Patrick Vallance gave the grim warning after more than 40 countries banned UK flights because of fears regarding the spread of the coronavirus variant, which is believed to be about 70 per cent more infectious.
France shut its border with the UK for 48 hours on Sunday night over fears of the mutant strain spreading further.
Boris Johnson has been in talks with the French president Emmanuel Macron after he announced strict Level 4 measures for parts of the UK on Saturday during an emergency conference, while First Minister Nicola Sturgeon warned “there is not a moment to lose on this”.
It has been reported plans to reopen the border will come into effect from Wednesday.
Sturgeon confirmed on Monday night she had spent most of the day in meetings, discussing the impact of the closure to accompanied freight.
She tweeted: “The most urgent priority is for UK to reach agreement with France on a protocol that will get freight moving again. There is not a moment to lose on this and while the UK government has lead responsibility here, we will do all we can to support.
“Our most immediate concern is for our exporting seafood sector. This is the peak time of year for business and the impact on them is significant.
“I raised the issue of compensation at the COBRA meeting earlier and the Scottish Government will also liaise with sector about support we can offer.
“We have no immediate concern about food supplies – supermarkets are well stocked – so I’d urge people not to buy any more than already planned. Obviously we will continue to monitor situation closely as we head into the post Christmas period. But early resolution vital.
“Similarly, we have good stockpiles of medicines and medical supplies, and no concern about Covid vaccine supplies – but again we will monitor the situation very closely.”
The Scottish Government “will be actively engaged in all of these issues until resolved,” she added.
However, she said “the most important priority is to reach agreement to get freight moving” as soon as possible.
Lorry drivers waiting to cross the Channel have been offered just a single cereal bar each, according to a trade association.
Rod McKenzie, managing director of the Road Haulage Association, told BBC Breakfast: "Clearly those drivers have spent a second night parked up somewhere, possibly on a motorway, possibly somewhere else, trying to get across the Channel, and many of them are European drivers trying to get home for Christmas, and their morale is very poor.
"Yesterday Kent County Council offered each of them one cereal bar, which is a pretty poor effort, I think in terms of maintaining their morale, and their spirits."
Mr McKenzie said toilet facilities were also a "big issue" with concerns over health and cleanliness.
He added: "We are not treating them well as a country, we are not treating lorry drivers well in these very difficult conditions that they are in at the moment."