The Peabody Trust, one of the UK’s oldest and best known housing associations, has been accused of failing to rehouse 19 tenants it plans to evict to make way for a new development.

Their plight highlights an acute shortage of affordable homes for low-wage key workers amid concern that too much new building is being targeted at more lucrative markets.

The tenants in five households, including 12 children and three NHS nurses, were all given what were supposed to be temporary housing on Peabody’s St John’s Hill estate in Clapham, south-west London, which is being demolished and replaced. Their rents were discounted to reflect the poor conditions of the homes, but are still more than twice the amount paid by their social housing neighbours.

Three of the families have lived in damp and overcrowded conditions on the 1930s estate for between seven and 13 years.

As they are tenants on short lets, Peabody has no legal duty to rehouse them in social housing, unlike their social housing neighbours who have been offered homes on the redeveloped estate.

Valeria Jemwa, a senior respiratory staff nurse and single mother with four children, is facing eviction from her Peabody flat.

One of the threatened tenants, Valeria Jemwa, a senior respiratory nurse and single mother with four children, said: “I have been in my flat for 13 years and they [Peabody] are making me homeless. They should feel a moral responsibility to help us, even if they don’t have a legal one.”

She asked: “Why am I being treated like a second-class citizen? Why don’t I deserve a replacement home on the new estate like the others? It just feels that nobody really cares.”

Speaking to the Guardian during a break from a shift at the Royal Brompton hospital, Jemwa said she and her children have been told to move out of their two-bedroom flat by 19 July. “I will wait until the bailiffs come to kick us out because we don’t have anywhere else to go,” she said.

Like the other four households also facing eviction, two as soon as next week, Jemwa has been offered other homes by Peabody but only on unaffordable rents and in areas several miles from her children’s schools. Despite qualifying for housing benefit and child tax credit she already struggles to pay her existing rent on her part-time salary.

Rent of £985 a month for one of the homes she was offered would leave Jemwa with just £552 from her net monthly salary.

“Peabody want to push me into somewhere they know I can’t afford,” she said.

Under Peabody’s own published policy, applicants for its market rent homes “must be able to afford the rent without further assistance”.

Jemwa and the other tenants are being advised by retired housing consultant Tony Bird, who has written good practice guides on “decanting” tenants on estates earmarked for demolition and previously trained Peabody staff on the issue. He asked: “Do Peabody really want to evict such wonderful public servants?”

The trust’s chair, Sir Bob Kerslake, the former head of the civil service, has been repeatedly asked by Bird to intervene, but each time Kerslake has referred the issue to their local council, Wandsworth, which has nominations rights for all new social housing in the borough.

Wandsworth’s new Labour cabinet member for housing, Aydin Dikerdem, urged Peabody to take responsibility for the families.

He said: “My priority is to work with Peabody to find suitable accommodation for these families so they can live where they need to be for work and their children. The housing allocation system has to be fair to the thousands of homeless families on our waiting list. But housing associations are allowed to use their discretion to solve emergency issues. Some of these tenants have been in Peabody accommodation for years and are precisely the kind of people in need that Peabody was set up to house.”

Dikerdem said the redeveloped estate included too many small homes for shared ownership and market rent. “Any regeneration that makes families like these homeless is not doing what it should be. Building market rate and shared ownership should not be made at the expense of homes for residents like these.”

In a statement Peabody said: “In an ideal world, all keyworkers and people on low incomes would have access to social housing. However, the 10 highest priority households on our internal rehousing lists include domestic abuse survivors, people fleeing hate crime, and people with severe medical disabilities. We could not prioritise intermediate rent tenants ahead of people with the highest level of need. To do so would be deeply unfair and a breach of our obligations to our social housing tenants.

“We are using discretion by offering alternative affordable homes to families and residents who do not have social rented tenancies. We’re also offering payments of almost £10,000 as well as other support and advice. They have all been offered homes that are significantly below market rent levels as close to the estate as possible. We’ll keep working with the council to try and find a solution that supports the families.”