The group expect the route to reopen later today.
Under current schedules, the first northbound train to Scotland is expected to be the 1030 from London, with the first southbound departure expected to be the 1340 from Glasgow.
Due to the continued uncertainty over the weather, ticket restrictions for passengers travelling to or from Scotland have been relaxed.
Customers with tickets to or from Edinburgh / Glasgow from Wednesday February 28 to Sunday March 4 can travel on any of our Scotland services during that time.
The company have urged customers travelling to plan ahead and check the website for details of trains before going to the station.
Refunds are also being given free of charge for those who wish to cancel their journey.