Nigel Farage hailed sweeping election wins for Reform UK as a “historic shift in British politics” on a day when the populist party made gains at the expense of Labour and the Conservatives.

Reform made advances in heartland areas of both parties, clocking up substantial early results in the English local elections by taking control of Essex county council, Havering – its first London local authority – and Sunderland city council.

However, the results were not without setbacks, for example in Harlow, a past general election bellwether, while one prominent pollster suggested the party may have peaked and that Farage would have reason to be “privately worried”.

Nevertheless, Reform established beachheads for the next general election in areas including Essex, home to the seats of prominent Tory MPs including Kemi Badenoch and James Cleverly, where Farage’s party went from having a single representative to taking control of the council.

“It’s a big, big day, not just for our party but for a complete reshaping of British politics in every way,” Farage said as he appeared on Friday outside Havering town hall, in a borough on the eastern border of Greater London where many voters identify more closely with neighbouring Essex.

Farage said the party was “two-thirds” of the way to where it wanted to be for the general election when it came to planning and fundraising. The door was now closed to Tory defectors, he said, but “the time is now” for conversations with “patriotic old Labour” MPs.

Reform’s first major gain was Essex, a key target for Farage’s party and a county council where the Tories had enjoyed majority control since 2001. Fifty-three Reform councillors were elected, a smaller majority than some had tipped, and will face a 24-strong opposition that includes 13 Tories.

Reform’s home affairs spokesperson, Zia Yusuf, said: “If that sort of result was replicated in the general election, Kemi Badenoch would lose her seat,” referring to the Tory leader’s constituency of North West Essex.

Suffolk county council was the second major local authority to go to Reform, whose 41 seats came largely at the expense of the Tories.

Reform’s capture of Sunderland city council, which had been Labour-controlled since 1973, was another seismic result. The party won 43 seatsin a city where a strong leave vote had been one of the turning points of the 2016 referendum on leaving the EU.

However, while the results showed Reform had gained support across swathes of the country, some targets eluded the party. The Tories secured all 11 district council seats available in Harlow, Essex. Reform also fell short of taking control of Bexley council, one of its targets in London.

Reform made a breakthrough in the elections to the Welsh Senedd, finishing second with 34 seats. Plaid Cymru was the largest party but six seats short of a majority, while Labour won just nine and finished third.

The elections expert Peter Kellner said that while Reform was likely to end up with gains of more than 1,000 seats overall, there were some warning signs. “Reform may well be on course for a record number of gains by any party in any elections, and in normal circumstances this would be astonishing. But if you compare it with a year ago it seems both in terms of seats and vote shares they are not going to do as well,” he said.

Reform won 41% of all seats contested across England in last year’s local elections, while this year’s tally appeared by Friday afternoon to be about 35%. “We have that record on recent polls and elections, and it seems clear that Reform has peaked,” Kellner said.

Any further slips in support would make a general election majority on this share of the vote much more difficult.

Reform’s vote share in English council seats so far has grown the most in areas with greater socioeconomic deprivation, early analysis by the Guardian showed. Figures from 691 wards show the party gained by an average of 20 percentage points in the least deprived areas and 30 percentage points in the most deprived areas.

Asked on LBC radio if Reform could form a government without winning big cities and metropolitan areas, Farage said: “Oh, yes … every party has areas where it’s weak and where it’s strong.” He insisted his party was competitive across vast geographical swathes of the UK.

At a rally on Friday night, Farage said his most important piece of advice to incoming Essex councillors was to keep disagreements behind closed doors.

“Please heed my advice. There are times when it’s better to bite your tongue, go and sit behind a desk, close the door, argue it out. But after four and a half years of psychodrama, with the last Conservative government, with Labour about to start ripping themselves apart, that’s the last thing of the voting public want to see. So, please, please, bear that in mind.”

Additional reporting by Isaaq Tomkins