There are delays of up to 20 minutes on First Great Western and Crosscountry services between Bristol Temple Meads and Taunton due to problem with line-side equipment between Weston-super-Mare and Taunton.
There is no estimate for normal service.
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The Duchess of Cambridge has been on maternity leave since the week before Easter.
The Duchess of Cambridge has been driven to St Mary's Hospital in London in the "early stages of labour", Kensington Palace has said.
Dry, breezy and cool