Britain’s housing crisis has hit a new low with not one of the single rooms available for private rent in large parts of London and Greater Manchester within the budget of people on housing benefit.

None of 87 rooms for rent in outer south-west London – which includes areas such as Feltham and Hanworth – were affordable for people relying on local housing allowance (LHA) and neither were any of the rooms in the southern Greater Manchester area, including Stockport and Wythenshawe, according to analysis of official 2018 data by London Councils, the local government association for the capital.

LHA is relied upon by 1.2m households to rent private accommodation but it has been frozen since 2016 and will continue to be frozen until at least 2020 as part of welfare cuts.

Council bosses have said that with private rents continuing to rise, sharp falls in affordability over the last three years are causing homelessness and child neglect as families cut back on essentials including food to make up rent shortfalls.

In Swindon and Newbury only 2% of one-bedroom flats available on the private rental market are now affordable, and in Northampton just 3%. Fewer than one in 10 three-bedroom family homes are affordable in Ipswich, Milton Keynes, Rugby, Luton and Cambridge.

In the capital, the plunge in affordability has been greatest in outer north-east London, which includes Ilford, Dagenham and Romford. In 2016, 30% of one- and two-bedroom homes were affordable, but fewer than 2% are now

Londoners on housing allowance were typically finding themselves £50 a week short and unable to pay their rent, London Councils said. Barking and Dagenham council reported a 200% rise in the number of people approaching them for help with housing over the last year.

Muhammed Butt, London Councils’ executive member for welfare, said the housing benefit freeze was “fuelling London’s skyrocketing rates of homelessness”.

“Pressures on household finances are immense and are a crucial factor in the increase in homelessness,” he said, adding that the number of homeless households in the capital was 50% higher in 2018 than at the start of the decade.

Mark Fowler, the director of community solutions at Barking and Dagenham council, said: “This is leaving people in a more precarious position. It means less money for food and clothing. You can see an increase in the neglect of children.”

The work and pensions secretary, Amber Rudd, told parliament in March she had concerns about the impact of the housing allowance on affordability and would discuss it with the chancellor before the budget, but the freeze remains in place.

In outer east London, which includes areas such as Walthamstow and West Ham, a quarter of rents were affordable for people relying on benefit looking for a one-bed place in 2016, but that has since fallen to less than 2%.

The median cost of a two-bedroom flat in outer east London, which includes Barking, is £316 per week. The maximum amount payable by housing benefit is £244. In Bristol and Cambridge, only 5% of the two-bedroom flats were affordable, and in Stevenage and central Northamptonshire it was just 4%.

“There is a gap between LHA and the bottom third of rents in 97% of areas across the country,” said Greg Beales, the campaign director at Shelter. “When housing benefit is so low that people are having to find over £50 of week to cover even the lowest rents, they face grim decisions between food, electric bills and keeping a roof over their head.”

Shelter, the housing and homelessness charity, is calling on the government to lift the freeze to stop more people falling into poverty and homelessness. It acknowledged the government had provided up to 3% extra for about 140,000 households hit hardest by the LHA rate freeze, but said this was not enough.

A government spokesperson said: “Providing quality and fair social housing is an absolute priority. The government increased more than 360 local housing allowance rates this year, by targeting extra funding at low-income households. We are also investing £4.8bn to build more affordable properties in London and have abolished the housing revenue account borrowing cap, giving councils across the country the tools they need to deliver a new generation of affordable housing.”