GWR said services running from South Wales through Bristol Parkway to London are suspended indefinitely, as are services between Bristol Temple Meads, Swindon and London.
The company said its staff were “working hard” to reopen the lines.
“Disruption is expected to continue for the rest of the day,” GWR said in a statement on X.
The line between Westbury and Weymouth was also closed due to a fallen tree, blocking the line at Yeovil Pen Mill.
Businesses including Longleat Safari Park near Warminster and Cotswold Park Farm in Cheltenham closed on Sunday due to the yellow weather warnings.
Meanwhile, both Wiltshire Police and Gloucestershire Constabulary urged people to travel only if necessary.
The A3102 between Lyneham and Tockenham in Wiltshire was closed on Sunday afternoon due to flooding.
It comes as properties are without power and motorists face travel disruption across the region.
In Somerset, the A3088 near Yeovil was closed from the junction for Montacute to the Garadore roundabout, after a five-vehicle collision – which included an overturned tanker.
Cineworld in Shaw Ridge, Swindon, said it had to “temporarily close” due to the impact of Storm Bert.
“Any customers whose screenings were cancelled will be automatically refunded,” bosses said in a statement, apologising for the inconvenience.
Avon and Somerset Police said officers were called to the crash at about 06:30 GMT on Sunday and two people have been taken to hospital with injuries that are not thought to be life-threatening or life-changing.
In Lydney in Gloucestershire, Newerne Street was closed on Sunday afternoon due to flooding.
According to travelling monitoring site Inrix, in Gloucestershire, the A4136 is closed in both directions due to flooding between the Yew Tree and Old Hill.
The M48 Severn Bridge was also closed in both directions due to strong winds, but has since reopened.
In Longhope in the Forest of Dean, the A48 was closed in both directions on Sunday due to flooding.
Due to the significant risk from flooding, Lydney Practice will be closed on Monday.
The surgery will contact patients who had booked appointments to make alternate arrangements.
Both the Quays Ice Rink and Hempsted Household Recycling Centre are closed in Gloucester along with the Flight Before Christmas grotto event at Cotswold Airport.
A number of trees have been blown down in Pittville Park in Cheltenham.
First Bus West of England said services across Bristol and the West “may experience delays, last-minute cancellations or diversions”.
Henbury Ford in Bristol is closed due to flooding, and bus services 1 and 2 are being diverted via Crow Lane, Old Crow Roundabout and Rugby Club.
Bus service 3 is being diverted along Station Road.
In Wiltshire, according to Scottish and Southern Electricity Networks (SSEN), about 1,900 people are without power.
Wiltshire Police is urging residents to only call 999 in an emergency, and is signposting those who wish to report obstructions and debris on roads to local authorities or National Highways.
“We are currently receiving a high number of calls related to the weather such as fallen trees or debris in the road,” it said.
“Please only travel when necessary. Be careful on the roads and adjust speeds and braking distances accordingly,” it added.
The force also said it was awaiting a tanker to clear the A350, which is closed in both directions between Bumpers Farm roundabout and Cepen Way South roundabout due to extensive flooding.
Road users are being told to find alternative routes.
Longleat Safari Park said it will reopen on Monday after having to close for safety reasons over the weekend.
Guests who had booked to see the Festival of Light will get refunded, the attraction said.
Scott Ashman, head of operations, said: “Our teams have been working hard to clear roads and train track, rebuilding fences as well as securing and repairing the lanterns for The Festival of Light.
“Refunds are still being processed and this will continue as a priority over the next couple of days.”
There is also reported flooding in the Malmesbury, Crudwell, Norton and Sherston areas.
In Swindon, Dorset and Wiltshire Fire Service said Maxwell Street and Lorne Street had been closed until further notice due to an unsafe building.
“Please avoid the area,” it added.
Meanwhile, Somerset Council said their teams had been “busy overnight responding to reports” of more than 30 fallen trees and areas of flooding.
“Please bear with us today as we’re dealing with a high call volumes and will be prioritising main roads,” it said.
It also encouraged motorists not to drive through flood water and to take care if travelling.