Must be a BBC thing; I frankly had absolutely no notion it wasn’t a customary utterance 
I work at a TV station behind the scenes.
The anchors generally have to be at work 2 to 4 hours before they appear on air, so they have plenty of time to run late.
It’s also not unusual to have several people who live near the station, so it’s always possible that if Anchor X cannot drive, Reporter/Anchor Y can walk.
Reporters and camera people spend a lot of time on the road, and so while they don’t have any special training, they do acquire a lot of experience driving. I estimate they average 3 hours driving in an 8-hour shift.
Finally, remember the first rule about TV: if the audience doesn’t know what was supposed to happen, it was a clean show. We’re not going to tell you someone was snowed in, or a reporter drove into three cars and so couldn’t be on air. We just go with our backup plan.
As everyone has said, even on snow days when the roads are covered with snow and ice the roads aren’t literally impassable. They’re just a lot tougher, and people who don’t absolutely positively need to be at work should just stay the fuck home so they don’t cause accidents that will make things worse. If you can take a day off, then stay home. If you can’t then by all means come to work, just budget a lot more time, have the right vehicle and equipment, and drive like you know what you’re doing.
Yup it’s generally OK for a few well-prepared people to travel in all but the most extreme conditions. However, even a small fraction of the normal traffic will cause complete gridlock and guaranteed accidents, which prevents the snowplows from doing their job, which makes the roads worse for everyone.
If (say) 99% of people stay home, the 1% who have to travel can do so reasonably well if they have an appropriate car, and skills. If instead 10% try to drive anywhere when they should be staying home, nobody can get anywhere.
Also remember that the places likely to need critical employees (hospitals, airports, police stations, etc.) are critical for others to get to, so routes to them are often the first routes plowed open. Even in a snowstorm, it’s easier to get to the hospital than your average manufacturing plant or office building.