Theresa May and her home secretary, Amber Rudd, have come under further pressure over the impact on the Windrush generation of the government’s immigration policy.

The former cabinet minister Sayeeda Warsi made it clear it was the policy itself that was at fault, rather than the way officials had implemented it, while senior Labour figures called on Rudd to resign.

Speaking on the BBC’s Andrew Marr Show, the shadow foreign secretary, Emily Thornberry, said it was hard to imagine a worse outcome from the policy brought in through a series of new laws from 2010 to create a hostile environment towards illegal immigrants.

Theresa May to blame for Windrush crisis, says Jeremy Corbyn Read more

“How much worse can it get? People have died, they have lost their jobs, and people working in the National Health Service all their lives are suddenly not even entitled to go to the National Health Service,” she said. “It couldn’t be worse and yet the home secretary thinks ‘I can apologise and it will be alright’. Well, it won’t be.”

The shadow chancellor, John McDonnell, backed Thornberry’s position. “If we are going to restore any sense of integrity to politics, they have to resign,” he said. “The home secretary now should accept her responsibilities, just as Theresa May said when we were in power.”

The prime minister has apologised and pledged to pay compensation in an attempt to end the row over the injustices meted out to children of Windrush-era migrants who had been unable to produce the documents demanded by the Home Office to establish their right to remain.

Play Video 0:26 Windrush generation to get compensation, says Theresa May – video

She and Rudd have, however, repeatedly blamed officials for the way the implementation of a policy they insist was the right one.

James Brokenshire, an immigration minister between 2014 and 2016, told ITV’s Peston on Sunday the situation was heartbreaking, but that as a minister he had consulted widely and made changes to the way the policy was to work.

“We put together a panel of experts, bringing together people from the equalities sector, homeless charities, local authorities to help support work around this and actually made changes to the way some of the things were done,” he said.

Peston on Sunday (@pestononsunday) #Peston asks @JBrokenshire about the @guardian's leaked letter showing a

Windrush case raised with him as Immigration Minister back in May 2016 pic.twitter.com/SZcmz6nmJh

Warsi, a former Conservative party chair and cabinet minister, said on the same programme that it was a failed policy caused by the party’s obsession with bringing down net migration.

Have you been affected by the Windrush deportation crisis? Read more

“I think we were all responsible. I would hold myself responsible as part of the government,” she said.

“What happened unfortunately during those years and has continued is that we had an unhealthy obsession with numbers. We were wedded to unrealistic targets, targets that we still haven’t met unfortunately a decade on – and yet we continue to remain wedded to targets.

“And what we ended up with was, I think, the unintended consequences of the policy we are now implementing.”

Warsi’s remarks, which reinforce criticisms made before the weekend by the former deputy prime minister Nick Clegg, who chaired the cabinet subcommittee on immigration from 2010 to 2015, undermine Rudd’s insistence in her statement to the Commons last week on blaming officials for thinking about policy rather than people.

Conservatives are likely to draw attention to Labour’s own commitment in government to creating a hostile environment, a phrase first used by Alan Johnson when he was home secretary in the last year of the Gordon Brown government, but campaigners insist that cracking down on illegal immigration could be done with better border checks rather than internal policing of status alone.