Historically the Brits, and Western Europeans in general, have a very good record for protecting people and dealing with such situations.

There is also a very good reason for wanting people to stay with their vehicles. If everyone stays with their vehicle rescuers can go to each vehicle and move people out systematically while keeping a tight inventory on everyone so nobody gets missed or left behind.

If people wander off the authorities can't so easily be sure that the people are safe. Did they pop into a nearby bar? Are they presently warm and safe and drunk? Or did they take off cross-country where they can't be found? Each person will have to be tracked down to make sure they haven't collapsed into a snow bank. Finding people who wander off would be a huge job that could require dozens of responders. All of which are potentially risking injury themselves.

Initial response would be to get to each vehicle and make an inventory of people and emergency needs. If the people are reasonably safe where they are it is better to leave them there until a major effort can be mounted to get people out systematically. Extraction would be limited to those in vehicles that can't protect the passengers and people with serious medical issues. Healthy people in situations that are not immediately dangerous can expect to wait. Hundreds uncomfortable, for however long it takes, as long as everyone gets out alive is preferred over most people comfortable, but a few dead.