Driver finds the safest spot available and parks there, waits for highway officials to open the road back up. You should hear from the carrier within a reasonable amount of time after a closure starts, then you'll get updates as things develop.
Most common spots that close on this route are the Coquihalla between Hope and Merritt, Highway 1 going through Fraser Canyon between Yale and Lytton, Trans Canada through Rogers Pass between Revelstoke and Golden. You can keep an eye on current road conditions at the
BC Ministry of Transportation DriveBC website before your car ships and while it's moving.
Winter closures on Trans Canada usually add somewhere between 12 and 48 hours to when your car gets delivered. Professional carriers aren't going to try risky alternate routes through mountain passes that haven't been plowed. They wait. Waiting ends up being safer, more reliable, and honestly faster than trying to go around on Highway 3 or Highway 16, both of which add hundreds of extra kilometers and cross more mountain terrain that doesn't get maintained as consistently.
Your delivery date is more like an estimate based on normal driving, not some guaranteed appointment time. Carrier confirms when pickup happens and gives you an estimated window for delivery. If you absolutely need your car in Edmonton by a specific date because of work, travel, whatever commitments you've got, book your transport one to two weeks earlier than your actual deadline. That gives you a buffer for weather delays, mechanical problems, general unpredictability of driving through mountains in winter.