The Metropolitan Police referenced the most recent terrorist attack when a bomb partially exploded on a London Tube earlier this month and within seconds videos and images of the so-called "bucket bomb" taken on smartphones had emerged on social media.
"We are particularly concerned when we see people young and old using their mobiles to film scenes when they should be moving away from the danger. The recent incident in Parsons Green (west London) is a good example of this," said Met Police deputy assistant commissioner Lucy D'Orsi, national policing lead for protective security.
"Our research showed that many young people think filming would be a good thing to provide evidence for police. We must get them to understand that the priority must be their safety," she said.
The latest drive is part of the Met's original 'Run, Hide, Tell' advice, which warns people to first try to run away from an attack scene, alternatively find somewhere safe to hide and eventually alert authorities once they are safe.
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The counter-terrorism police network wants the new 'Run, Hide, Tell' message to be routinely taught in schools across Britain as part of the national curriculum within the citizenship module.
The Met said the message was all the more important as there have been an increasing number of young casualties in recent terror attacks, including seven children among the 22 killed in a suicide bombing at an Ariana Grande concert in Manchester in May.
John Cameron, head of helplines at the charity, encouraged adults to listen to children's worries and reassure them that terror attacks are rare.
"Although these conversations might be difficult, the spate of devastating events means that they cannot be brushed under the carpet and we all have a duty to help every child stay safe," he said.
Britain has faced five terrorist attacks this year and the terror threat level remains at 'Severe' indicating an attack is still highly likely.