Ex-Archbishop of Canterbury Lord Williams, Muslim and Jewish faith leaders are among those to sign the open letter to Theresa May.
It describes the camp as “a stain” on France’s and Britain’s conscience.
The UK government stressed it was committed to resettling vulnerable children.
Meanwhile, a Red Cross report, No Place for Children has said lone children with family in the UK are being left to fend for themselves because of bureaucracy.
French president Francois Hollande has said the Jungle will be cleared before winter, with its 9,000 inhabitants due to be dispersed around the country.
The letter to Mrs May was organised by the Citizens UK charity.
It tells Mrs May that the children, the youngest of whom is eight, “have fled conflict and persecution [and] are now stuck in northern France, deeply traumatised and at great risk, as well-documented by the anti-slavery commissioner you yourself appointed while home secretary“.
It adds: “None of us want the Calais ‘Jungle’ to exist. It is a stain on the conscience of both France and Britain.
“But in haste to clear it, the need to protect children is even more paramount”.
“During the last such demolition, the charity Help Refugees documented that 129 children went missing. The time to act is now.”
Other signatories to the letter, which echoes an appeal by Unicef last week, include Muslim Council of Britain secretary general Harun Rashid Khan and Laura Janner-Klausner, the senior rabbi to the Movement of Reform Judaism.
They are joined by the bishops of Durham, Manchester, Barking, Stepney and Southwark.
Charities estimate there are about 1,000 unaccompanied minors in the Jungle and about 400 could be resettled in the UK.
About 200 could be resettled under the EU’s Dublin regulation, which allows lone refugee children to be placed in a country where they have a relative who can be responsible for their care.
An amendment to the Immigration Act also requires the government to arrange for the transfer to the UK of unaccompanied refugee children from Europe.
The Red Cross report found failures at every point in the process of reuniting children from the camp with their relatives in the UK.
It said the process took about 11 months, often because of basic administrative errors and staff shortages in Calais, which left children disillusioned and desperate.
Three children, who had a legal right to join family in the UK, have died trying to make their own way to the UK, the charity said.
Alex Fraser, from the British Red Cross, said: “Children with a legal right to be here should be on the Eurostar across the channel, not being dispersed somewhere else in France, with no idea when they can join their family.”
A Home Office spokeswoman said the camp’s clearance was a matter for the French authorities, he added, that the UK had made “crystal clear its commitment to resettle vulnerable children“.
She added: “We continue to work with the French government and partner organisations to speed up mechanisms to identify, assess and transfer unaccompanied refugee children to the UK, where this is in their best interests.”