Flooding means the line is closed between Reading and Taunton and Swindon and Bristol Parkway.
GWR said: "To help customers make their journey, those with tickets for January 4 and January 5 are able to travel up to and including Saturday, January 6."
Services in the Cotswolds continue to operate but could be reduced due to flooding elsewhere.
Anyone affected by delays can claim compensation at gwr.com/delayrepay.
Customers who purchased tickets but do not travel can claim a full refund or amend their ticket.
Those who travel and are delayed could be entitled to compensation if they are delayed by 15 minutes or more.
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