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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The high of 35C was recorded on a Bakerloo line train between Piccadilly Circus and Oxford Circus at 2pm.
Weather forecasters say temperatures could reach 35C, beating the previous record of 34.6C in the scorching summer of 1976.
After years of political manoeuvring over the future expansion of Heathrow Airport, there has been an important development this lunchtime.