Operator First Great Western says season ticket holders will be compensated for the poor performance.
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Arrangements have been made for First Great Western rail tickets to be accepted for these journeys.
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Cross Country and First Great Western services between Oxford and Banbury have been disrupted.
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First Great Western is currently advising passengers not to travel between Plymouth and Bristol.
Railway operator First Great Western said services between London Paddington and south Wales were affected.
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National Rail said First Great Western was expected to run a full service on Saturday.
First Great Western and Chiltern Railways are running usual timetables, but are prepared for disruption.
First Great Western was operating a reduced timetable across some of its network due to the weather.
But railway operator First Great Western says it does not mean the route will be fully re-opened.
Transport for London services are carrying First Great Western passengers via any reasonable route "until further notice".
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Meanwhile First Great Western said passengers should check if their service is running before setting out.
First Great Western trains have been subject to a number of alterations and delays, with some routes cancelled.
Mark Hopwood, First Great Western managing director, said it was the "most extensive" phase of the upgrade yet.
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First Great Western said services running through these stations may be cancelled, delayed or revised at short notice.
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First Great Western's high-speed services from London Paddington to Wales are either being cancelled or diverted through Box.
Operator First Great Western advised customers not to attempt to travel west of Taunton in either direction.
The project is a partnership between First Great Western and Brompton Dock - a sister company of Brompton Bicycles.
The compensation has been paid to operators including First Great Western and CrossCountry.
First Great Western said train services from Cheltenham to London were delayed in the Swindon area due to flooding.
First Great Western said staff did not break any rules, but apologised, saying more discretion should have been shown.
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First Great Western said some services were being affected by snow with two trains from London to Bristol cancelled.
First Great Western (FGW) and the British Transport Police say fans risk missing the final if they are caught.
First Great Western has published a list of diverted routes effective from 23:00 on Saturday to 03:30 on Monday.
He said the force was working with First Great Western and Arriva Trains Wales, which operate services in south Wales.
Commuters travelling on First Great Western services have been experiencing delays of up to 90 minutes due to signal failures.
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Chiltern Railways, First Great Western, London Midland, Scotrail, Stansted Express and Virgin Trains are all subject to cancellations and alterations.
First Great Western operated a reduced service, running about half the normal number of trains into and out of Paddington.
Replacement buses will be supplied, and tickets will be accepted on First Great Western, Southern, London Underground and London Overground routes.
Services to Heathrow Airport were suspended and First Great Western trains were cancelled after a fire damaged signal equipment on Thursday.
First Great Western's managing director, Mark Hopwood, and former Labour MP for Stroud, David Drew, carried out the naming ceremony at Stroud station.
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