By Kylie MacLellan and Guy Faulconbridge

LONDON (Reuters) - Britain pledged on Monday to investigate claims that politicians may have sexually abused children in the 1980s in a conspiracy by members of the establishment who used their power to cover up the crimes.

The allegations have jarred the current political elite just as Britain is grappling with revelations that some national celebrities had sexually abused children for decades.

"We are going to leave no stone unturned to find out the truth about what happened," Prime Minister David Cameron told reporters.

"Three things need to happen: Robust inquiries that get to truth; police investigations that pursue the guilty and find out what has happened; and proper lessons learned so we make sure these things cannot happen again," he said.

Interior minister Theresa May told parliament an independent review would be held of a 2013 investigation carried out by the interior ministry into the handling of allegations of abuse by politicians.

The government will also establish an independent inquiry panel of experts to look more broadly at whether public bodies have taken seriously their duty of care to protect children from sexual abuse, May said.

That inquiry, which she said was unlikely to report back before a national election in May next year, will look at organisations including the BBC, the health service, religious authorities and political parties.

No evidence has yet been published to support claims that there was a paedophile conspiracy inside the political elite.

But the unmasking of late BBC television presenter Jimmy Savile as one of Britain's most prolific sex offenders has forced a wider questioning about how paedophiles in positions of power could sow such damage while evading detection for so long.

Once feted as a national treasure, Savile is now known to have used his fame to get unsupervised access to his victims, raping and abusing girls, boys, men, women and even dead bodies.

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Veteran entertainer Rolf Harris, a household name in his native Australia and adopted home Britain, was jailed for almost six years last Friday on 12 counts of assaulting four girls, some as young as seven or eight, between 1968 and 1986.

'ABUSE OF POWER'

Fears that claims of abuse by politicians were not properly investigated in the 1980s were stoked when one of former Prime Minister Margaret Thatcher's most trusted advisers admitted there may well have been a cover-up of child abuse at the time.

"At that time I think most people would have thought that the Establishment, the system, was to be protected," Norman Tebbit, a former Conservative minister, said on Sunday.

"And if a few things had gone wrong here and there that it was more important to protect the system than to delve too far," he said. "That view was wrong.

Such a blunt assessment of the priorities of an earlier Britain was explosive, prompting front-page headlines about "V.I.P. Paedos" in local print media and leading national television news bulletins.

Abuse claims have sullied the reputations of some of the world's most venerated institutions: Pope Francis on Monday told victims of sexual abuse by Roman Catholic clerics that the church should "weep and make reparation" for crimes.

In Britain, local media have alleged that a group of British politicians and others in positions of authority may have used their position to abuse children in state care during the 1980s. It was not possible to independently evaluate those claims.

British police in 2013 began an investigation known as Operation Fernbridge into allegations of child abuse in the early 1980s at the Elm Guest House in London.

For campaigners working with the victims of child abuse, the domestic political storm shows Britain may be finally preparing to face some of the demons from its past.

"People just didn’t want to talk about it in the past," Jon Bird, a 56-year-old who works for the National Association of People Abused in Childhood charity, said.

"You hope they are going to take it seriously and give it the resources and powers to actually get to the bottom of it," said Bird, who was raped when he was 4 years old. "So far it is just words."

(Additional reporting by William James; Editing by Jeremy Gaunt and Sonya Hepinstall)