Shadow education secretary Angela Rayner has claimed Boris Johnson is “in cahoots” with Nigel Farage, and urged voters in the Midlands and the north to stick with Labour, or risk the election of “the most hard-right government we’ve seen since Margaret Thatcher”.

Labour has seized on Farage’s announcement on Monday that Brexit party candidates would stand aside in Conservative-held seats to help deliver Johnson a majority.

“Nigel Farage is working in cahoots with Boris Johnson, trying to hoodwink the public – and they are very friendly with Donald Trump,” Rayner told the Guardian.

She claimed all three men want “a hard-right nasty Tory government, that will privatise and deregulate our markets – and they will make Margaret Thatcher look like a pussycat”.

Johnson welcomed Farage’s decision to stand down more than 300 candidates on Monday; but Downing Street has denied any pact between the two parties.

Labour strategists fear the Brexit party’s absence in more than 300 seats will make it harder for them to take the Tory-held marginals they must win to get Jeremy Corbyn into Downing Street, though they also hope the Brexit party will split the leave vote in some of those seats they need to hold.

Rayner acknowledged that many voters in constituencies like hers, Ashton-under-Lyne on Tameside, felt disappointed that Brexit has not yet happened.

“They feel let down, and they’ve got a right to feel let down. For three years, they were promised that Brexit will be done, and over two prime ministers they’ve not delivered it.”

She was speaking on Labour’s bright red battlebus after launching the party’s lifelong learning policy at Blackpool football club, alongside Jeremy Corbyn.

They made a brief campaign stop in Blackpool town centre, before the Labour leader was whisked off to Doncaster to visit flood-hit communities.

A small crowd of supporters gathered, some keen for selfies with Corbyn and Rayner – but other passersby appeared less than enthusiastic.

One young woman challenged a Labour aide about Corbyn’s stance on defence, asking how he had dared to lay a wreath at the Cenotaph on Sunday; a passing car slowed down so its passenger could make a series of insulting hand gestures; and a man on a mobility scooter insisted he couldn’t vote for Corbyn because he “doesn’t want Brexit”.

But Rayner said Labour’s Brexit policy was “pragmatic” and the threat of a rightwing Conservative government should give voters pause for thought.

“Anyone has got to put aside their Brexit differences, and see the challenge that faces us,” she said. “Everything that we built, the Labour party, from 1945: the welfare state, the National Health Service, the council housing that we have, the education system, is all at risk if Nigel Farage, Boris Johnson and Donald Trump get their hands on No 10 on 13 December.”

She said Labour’s headline-grabbing offer of six years’ free adult education – part of its promise of a universal “national education service” – would help people who miss out on education the first time around: as she did when she had her son aged 16.

“It unlocks potential. It deals with loneliness. It gives so much potential for people, and that’s why I’m so animated about it, because I know how vitally important it is to our communities to give that hope to people – that they can strive for whatever they want to achieve. That they can dream big, and they can do it,” she said.

Rayner also promised that a Labour government would take vocational qualifications as seriously as academic study.

Asked how she would achieve that when other politicians have failed, she said, “none of them get it, because none of them have lived it like I have”. She gave her son’s case as an example.

“My son is an enigma compared to a lot of politicians’ children, because he went through FE [further education]. He works in a nightclub now, and he’s dead happy. And all I wanted for my son was for him to be happy; to be in gainful employment that he enjoys doing, and to be able to provide for his family.”

She highlighted Labour’s plans to abolish testing in primary schools, as further evidence of her determination to transform the culture in education.

“The current testing and narrow curriculum that’s there at the moment is so much pressure on young people, it’s actively adding to the problem rather than trying to alleviate it. I don’t want to stick a sticking plaster on it, I don’t want to fix children once the system’s broken them, I want to give every child the opportunity before that – because the system should protect and nurture, and not damage our children,” she said.

The national polls continue to point to an overall majority for Johnson’s Tories: but with more than four weeks of the campaign still to go, Rayner insisted Labour is ready for government.

“There’s no question that we have a better programme for government and a better calibre of people to deliver it.”

She was enthusiastically received by Labour activists at Tuesday’s press conference, and is widely perceived as a candidate to replace Tom Watson as the party’s deputy leader – or perhaps to succeed Corbyn himself, if the general election result goes against him.

Shadow women’s minister Dawn Butler has already declared her candidacy for the deputy leadership; but other likely contenders, including Laura Pidcock and Rebecca Long-Bailey, have declined to say whether they will stand.

Rayner also refused to be drawn on her future. “We’ll have to see, when that position arises. But at the moment we’ve got to get Jeremy Corbyn into No 10.”