Wandsworth council has 99 blocks of 10 storeys or more - more than any borough in London - and following the Grenfell fire last June set aside £30 million of Council resources to pay for vital safety measures such as replacing cladding and installing sprinklers.
Some 71 people died in the blaze on June 14 past year.
But according to a source briefed on the findings, her report stops short of calling for significant changes to the existing regulations.
Dame Judith Hackitt has called for "radical reform" of fire safety rules, saying they are not now fit for objective.
Dame Judith Hackitt, who led the review, said that "deep flaws" have been identified in the current system, far beyond the question of the specification of cladding systems.
"It is clear from Grenfell and from the other tests that have been done that despite the guidance now saying that, people were putting other materials up, so I don't think a ban will work". She said the problem was that people were cutting corners and ignoring the regulations. The Local Government Association, which represents councils, said: "It is great that the government has honoured its commitment from last summer to meet the unexpected exceptional costs for councils". An aluminium cladding with a combustible plastic core is thought to have contributed to the rapid spread of the fire.
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During Prime Minister's Questions this week Theresa May told MPs: "I can today confirm that the Government will fully fund the removal and replacement of risky cladding by councils and housing associations, estimated at £400m".
Dame Judith, an engineer who chaired the Health and Safety Executive for almost a decade, told Today that she "absolutely recognised the level of concern and distress" felt by Grenfell survivors and residents of similar tower blocks around the country.
The Royal Institute of British Architects has campaigned for a ban on combustible cladding and says above all its members need clarity.
"Cladding is one issue".
"This illustrates the siloed thinking that is part of the problem we must address".
Speaking in the House of Commons three hours after the publication of the review, Mr Brokenshire said new laws would be introduced that delivered "meaningful and lasting" change to the building safety system.
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Such bills allow Congress , with a majority vote in each chamber and the president's signature, to overturn recent agency moves. Neither tipped a hand until they voted a few hours earlier Wednesday to move the measure past a procedural hurdle.
Shahin Sadafi, chair of the Grenfell United survivors group, told Today it was "very disappointing" that the report appears not to include an outright ban on flammable cladding.
The report also falls short on recommending a complete ban on the use of desktop studies, despite recent research revealing their "utter inadequacy".
"What keeps us awake at night is knowing that there are schools, hospitals and homes that have this cladding on them".
Housing Secretary Rt Hon James Brokenshire MP said: "People must always feel safe in their own home".
"The simple solution that doesn't risk more mistakes is to only use non-combustible cladding", he said.
He added: "This is an industry that has put profit above people's safety".
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