The prime minister is expected to continue his reshuffle on Thursday, concentrating on the lower ministerial ranks, with vaccines minister and schools minister among the roles yet to be filled.
Robert Buckland and Robert Jenrick departed their roles as justice secretary and housing, communities and local government secretary, respectively.
Schools minister Nick Gibb was also shown the door.
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Priti Patel kept her role as home secretary, despite speculation that she would be sacked, while Rishi Sunak will continue as chancellor.
In a tweet following the reshuffle, the prime minister said: “The cabinet I have appointed today will work tirelessly to unite and level up the whole country.”
Mr Williamson was targeted after his mis-handling of schools during the coronavirus pandemic, including closures and a fiasco over the awarding of A-level and GCSE grades.
Mr Zahawi, on the other hand, was rewarded for his efforts in the successful COVID-19 vaccine rollout – almost 90% of those over 16 have had a first dose of the vaccine, while just over 81% are fully vaccinated.
Following his appointment as education secretary, Mr Zahawi said: “Children and young people have had a tough time during this pandemic and I’ll be listening to them and their families as we accelerate our work to build back better and fairer.
“From my own experience, I know what a beacon of opportunity this country can be and I want all children, young people and adults to have access to a brilliant education, the right qualifications and opportunities to secure good jobs.”
Oliver Dowden was made co-chairman of the Conservative Party, replacing Amanda Milling.
Anne-Marie Trevelyan, who was international development secretary before her department was merged with the Foreign Office last year, returned to the fold as trade secretary.
Nadine Dorries took Mr Dowden’s culture secretary job.
Steve Barclay succeeded Michael Gove as Chancellor of the Duchy of Lancaster, while Mr Gove replaced Mr Jenrick at the Ministry of Housing, Communities and Local Government, also taking on cross-government responsibility for Mr Johnson’s “levelling up” agenda.
Health Secretary Sajid Javid, Work and Pensions Secretary Therese Coffey, Commons Leader Jacob Rees-Mogg, and Northern Ireland Secretary Brandon Lewis all kept their jobs.
There will also be a rise in maximum maintenance loans to increase in line with inflation, giving an increase of £414 a year to help students with living costs.
However, the education secretary did not say if the rise would continue after that.
“We’re going to look at this and the maintenance support and the sector overall as part of the reform that we intend to set out in the months to come,” she said.
“So no decision, no decision has been taken on what happens beyond this.”
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She said the government will be looking at “what is required… to get our universities on a more sustainable footing… but also to deliver a better deal for students as a part of that”.
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0:57
University tuition fees to increase
The minister said she also “intends to look at” uprating the threshold at which students need to start paying tuition fees back in line with inflation.
Jo Grady, general secretary of the University and College Union (UCU), said the tuition fee rise was “economically and morally wrong”.
She said: “Taking more money from debt-ridden students and handing it to overpaid underperforming vice-chancellors is ill conceived and won’t come close to addressing the sector’s core issues.”
The National Union of Students (NUS) said students were being asked to “foot the bill” to keep the lights and heating on in their universities and to prevent their courses from closing down amid the “crisis”.
Alex Stanley, vice president for higher education of the NUS, said: “This is, and can only ever be, a sticking plaster.
“Universities cannot continue to be funded by an ever-increasing burden of debt on students.”
Universities have been making up for fees being frozen since 2017/18 by taking in international students who pay more.
However, student visa numbers have fallen after the previous government made it more difficult for them to come to the UK recently, so universities can no longer rely on the fees.
On Monday afternoon, the two biggest jobs were confirmed, with former home secretary Ms Patel being given the shadow foreign secretary role.
Former shadow work and pensions secretary Mel Stride, who ran in the Tory leadership race and is considered more of a moderate than Ms Badenoch, has been made shadow chancellor.
Robert Jenrick, who lost out to Ms Badenoch, is the new shadow justice secretary, sources told Sky News.
Earlier in the day, Laura Trott, who served as chief secretary to the Treasury under Rishi Sunak, was appointed shadow education secretary.
The new Tory leader made her first appointments on Sunday evening ahead of her new top team meeting for the first time on Tuesday.
Now the Conservatives are in opposition, the shadow cabinet’s role is to scrutinise the policies and actions of the government and to offer alternative policies.
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Nigel Huddleston and Dominic Johnson, junior ministers under Mr Sunak, were appointed joint chairmen of the Conservative Party.
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1:25
Badenoch: ‘We let standards slip’
Former foreign secretary James Cleverly, who came third in the leadership race, said on Friday he would not be joining Ms Badenoch’s top team.
Ex-prime minister Mr Sunak, his former deputy Sir Oliver Dowden, ex-chancellor Jeremy Hunt and former Brexit, health, and environment secretary Steve Barclay have all said they will be joining him on the backbenches.