TransPennine Express has also cancelled Monday services between Carlisle and Glasgow Central and Edinburgh in both directions, as well as services between Newcastle and Edinburgh.
The firm said tickets dated for Monday could be used to travel on Tuesday or a refund could be sought.
LNER said tickets dated for Monday would now be valid for between Sunday and Wednesday.
Avanti West Coast, which stops at Oxenholme, Penrith and Carlisle, said Monday tickets for travel north of Preston would be accepted between Sunday and Tuesday.
Storm Floris is the sixth named storm of the 2024-25 season, and the first since January.
There is also a yellow warning across northern England, north Wales and Northern Ireland from 06:00 BST on Monday through to midnight.
