Millions of struggling low-income households will not benefit from the £2bn “tax cut for the low paid” announced by the chancellor in the budget as a way of easing the pain of soaring living costs, according to campaigners.

The reduction from 63p in the pound to 55p in the universal credit taper rate – the amount in benefits a claimant loses for each pound they earn above a set work allowance – was intended to soften the blow of the withdrawal this month of the £20-a-week universal credit uplift.

Calling it a “tax cut for low-paid families”, the Rishi Sunak said about 2m qualifying households would be at least £1,000 a year better off. He also announced a £500 boost to the work allowance and confirmed a 6.6% rise in the “national living wage” to £9.50 an hour.

There has been increasing concern, including on the Tory backbenches, that the removal of the £20-a-week uplift will hit millions of households already struggling with rent, soaring energy and food bills and next April’s rise in national insurance contributions.

The Institute for Fiscal Studies (IFS) said the changes meant a full-time minimum wage worker on universal credit would have their disposable income increase by £250 a year as a result of the minimum wage rise and an extra £1,000 a year or more as a result of the universal credit changes.

However, the shadow work and pensions secretary, Jonathan Reynolds, said: “While it’s welcome that the chancellor is following our lead and reducing the taper rate, he is taking people for fools if he thinks this alone makes up for the biggest ever cut to social security, tax hikes and a cost of living crisis.”

Anti-poverty campaigners said it provided no extra support for millions of people who were unemployed or unable to work because of disability or illness, or claiming working tax credits. The boost amounted to just a third of the £6bn a year spent on the universal credit uplift received by 5.5 million claimant families.

Katie Schmuecker, the deputy director of policy at the Joseph Rowntree Foundation, said: “The reality is that millions of people who are unable to work or looking for work will not benefit from these changes. The chancellor’s decision to ignore them today as the cost of living rises risks deepening poverty among this group, who now have the lowest main rate of out-of-work support in real terms since around 1990.”

The chancellor’s changes reset the taper to the more generous level envisaged by the political architect of universal credit, Iain Duncan Smith, as a way of improving work incentives. He was overruled by George Osborne, who as chancellor set it instead at 65%. It was reduced to 63% in 2017.

The taper rate cut – more generous than a mooted 60p cut – may placate nervous Tory backbenchers: in 191 Tory-held parliamentary seats more than 30% of low-income working families were hit by the cut to the universal credit uplift. Labour has previously proposed reducing the taper but did not say at what rate it should be set.

Sara Ogilvie, the policy director of Child Poverty Action Group, said: “The long-overdue decision to lower the universal credit taper rate will help lots of low earners. But there was nothing for those who cannot work – carers, those with young children and people who are sick or disabled – who face the same cost pressures as other households and will still have a black hole in their finances after the universal credit cut.”

Louise Rubin, the head of policy at disability equality charity Scope, said: “The budget does little to address the fears of many disabled people and families about their finances this winter. The cut to universal credit, sky-high inflation and spiralling energy costs will hit disabled households hardest.”