Kemi Badenoch has said it would be wrong to think a change of leader would “fix everything” after the Conservatives suffered heavy losses in English local elections.

The Tory leader told the BBC her party would “come out fighting” after losing 674 council seats and control of 16 authorities.

But she cautioned against a knee-jerk response, saying the party’s issues were “not going to be fixed after six months” of her leadership.

She said: “Reform had a good night. We had a bad night. And what this shows for a lot of people who hoped that just changing leader again would fix everything is that that’s not going to be enough. We tried that previously. And that brought us to a historic defeat.”

Speaking to the BBC’s Sunday with Laura Kuenssberg programme, Badenoch said protest was “in the air” in these local elections and argued her party was working on a “slow and steady” plan to return to government by the end of the decade.

She said: “We are going to come out with the policies that people want to see, but what we are not going to do is rush out and tell the public things that are not true just so we can win votes.

“This is not about winning elections; this is about fixing our country. Yes, of course, you need to win elections to do that, but you also need a credible plan.”

Asked if it was feasible that Reform UK leader Nigel Farage could be the next prime minister, she said “anything is feasible”, noting Australian Prime Minister Anthony Albanese had won re-election this week after trailing in the polls.

But she added: “My job is to make sure that he [Farage] does not become prime minister because he does not have the answers to the problems the country is facing.”

Badenoch was a cabinet minister in the previous Conservative government and took over as party leader last November, following its worst defeat ever in a general election.

The former leader of North Northamptonshire council, Jason Smithers, told the BBC he would call for Badenoch to resign as leader following this week’s results.

But shadow justice secretary Robert Jenrick – a previous contender for the Tory leadership – has backed Badenoch, telling the BBC she was doing an “excellent job”.

The results in the local elections were worse than Conservatives had feared, with the party not only losing councils to Reform but also the Liberal Democrats.

It lost 674 council seats and control of all 16 local authorities it was defending but wrested the Cambridgeshire and Peterborough mayoralty from Labour.

Reform UK was the big winner, taking control of 10 councils and gaining 677 councillors, as the party seemed able to tap into widespread dissatisfaction with the Conservatives and Labour.

The Lib Dems were the other standout winners, gaining 163 seats and control of three councils. They seized Shropshire from the Tories and gained control of Oxfordshire and Cambridgeshire after both previously had no party in overall control.

The Green Party made some gains, winning 44 seats in total – however it suffered disappointment in the West of England mayoral race.

The BBC is estimating that, if elections had taken place across the UK on Thursday, the Conservatives would have slumped to just 15% of the national vote, its worst-ever share of such a projection, behind the Liberal Democrats on 17%.

Labour would have won 20% of the vote, according to the projection, equalling its lowest previous recorded performance in 2009.

Health Secretary Wes Streeting told Laura Kuenssberg Labour had “got the message” in these local elections and asked voters to “give us a bit of time” to change things.

When asked about Labour’s record in government since winning the last general election, Streeting said “there’s no doubt that there are some individual things we’ve done that people have not liked”.

He added: “Every single measure we have taken to raise more money for the NHS has been unpopular, has been contested.

But he claimed it was already showing results in terms of reduced waiting lists.

Liberal Democrat leader Sir Ed Davey said his party had achieved strong results across the country and had taken the title of “party of Middle England” from the Conservatives.

He said many people would be worried about Reform UK controlling councils and what Farage had said about cutting spending on climate change initiatives and special education needs.

Sir Ed said the Lib Dems were taking on Reform UK in many areas.

“If we can expose the fact that Nigel Farage is so keen on Donald Trump that will mean that the Liberal Democrats can come through,” he said.

The Green Party had also made big strides across the country, its co-leader Adrian Ramsay told the programme.

Speaking about Reform UK’s performance, he said the success of Farage’s party was not built on strong foundations.

“It remains to be seen whether they’re willing to put in the hard yards,” Ramsay said. “Green councillors have a strong track record in our communities.”

But asked why the Greens didn’t do as well as Reform UK in these elections, Ramsay said his party was still “growing” and urged voters to “be part of our movement” if they wanted a different kind of politics.