The damage was caused by a high-speed Class 802 Hitachi train which was being tested between London and Bristol ahead of its handover to GWR.
South Western Railway, Chiltern Railways, Virgin Trains, West Midlands Railway and Transport for Wales are all accepting GWR tickets, while tickets for GWR services today will also be valid tomorrow.
Network Rail said engineers worked overnight to make safe all four lines between Reading and Paddington.
On Thursday, most services are expected to operate as usual, but there'll be a reduced timetable on the London Paddington to Reading route, and delays are expected.
TRAIN services to and from London Paddington have been partially restored after damage to overhead wires prevented travel into the station this morning.
About 500m of overhead cables were "severely" damaged during a Hitachi train test run on Tuesday night. "We advise passengers to check before they travel with their train operator or visit www.nationalrail.co.uk for the latest travel information".
Ten of thousands of commuters were unable to get to work after all services in and out of London Paddington were cancelled.
It followed an evening of major delays as electricity supply problems caused delays and cancellations on a number of busy routes in and out of London.
Elsewhere, travellers trying to get from Slough to Windsor and trying to catch a train to Waterloo instead of Paddington, as advised, faced a further 35-minute delay at one point because there was no driver available for the short journey.