Campaigns manager Steve Hyndside, from Stroud, said he travelled more than two hours - while on holiday in Mid Wales - to the coronavirus test centre near Ironbridge, but was told it had run out of tests and closed after he arrived five minutes early.
There were reports of traffic queues and chaotic scenes at the centre, in Jiggers Bank at around 6pm yesterday, resulting in the site being shut by police.
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Broadcast journalist Piers Morgan responded by describing the testing process as "appalling".
Health Secretary Matt Hancock said there had been a surge in people seeking “inappropriate” coronavirus tests when they do not have symptoms of Covid-19.
Mr Hancock also described how some people who were going on holiday had sought to get tests, which is “not what the testing system is here for”.
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Speaking on Twitter, Mr Hyndside added he had two ill children, aged one and three, with him at the time.
He said: "Utterly shambolic – just drove over two hours with two ill children (one and three years old) to a test site in Telford for an appointment booked for 6.30pm.
"We arrived at 6.25pm to be told that the whole site was closed because they've run out of tests."
UTTERLY SHAMBOLIC - just drove over 2 hours with two ill children (1 and 3 years old) to a test site in Telford for an appointment booked for 6:30. We arrived at 6:25 to be told that the whole site was closed because they’ve run out of tests!— Steve Hyndside (@SteveHyndside) September 8, 2020
Piers Morgan responded to Mr Hyndside and described the testing process as "appalling".
In a tweet directed to Prime Minister Boris Johnson and Health Secretary Matt Hancock, Morgan said: "This is appalling. What the hell is going on with your ‘world class testing’."
Telford MP Lucy Allan said she was shocked to to hear that the centre had be closed after staff ran out of test kits and she was due to hold talks with Public Health England about what went wrong.
She posted on Twitter: "I'm shocked to hear reports that Telford testing centre closed last night after being overwhelmed with people travelling across country for a test after a computer error."
Several people took to social media to complain about the situation including a family on holiday in Mid Wales who drove for two hours with their two unwell young children, only to be turned away by staff.
Meanwhile, people in Shropshire say they have been asked to travel to Oldham in Greater Manchester, while others have been directed to Leicester and Builth Wells.
The Government website, used to book Covid-19 tests, has also been telling people that there are no walk-through test sites or home test kits available.
Additional reporting by Deborah Hardiman