Fast trains between Didcot and Oxford, and Paddington and Oxford are cancelled. Stopping services are running every 30 minutes.
Flooding between Swindon0Bristol Parkway is causing delays of up to 15 minutes between London Paddington and Cardiff Central. Updates soon.
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Mayor of London Sadiq Khan is backing the scheme.
A spokesperson said the lightning strikes had left the "aircraft fuelling system unavailable."
Nearly 1,000 properties have been left without power, and road users have been warned of potentially hazardous conditions.