LONDON — From her doorstep in the exclusive Knightsbridge district of London, Christine Cotter dismissed the idea of voting for Britain’s opposition Labour Party in elections on Thursday, describing its left-wing leader, Jeremy Corbyn, as a “dreamer” whose policies would bankrupt the country.

But in the rare spring sunshine, Ms. Cotter added that, even in her area of multimillion-dollar properties, not everyone shared her views about Labour.

“There is fear that they could win in Westminster,” said Ms. Cotter, referring to her affluent borough, which will be among those to hold local council elections on Thursday. She noted Mr. Corbyn’s following among younger voters: “He’s selling a dream,” said Ms. Cotter, 71, a semiretired psychotherapist.

In theory, the voting on Thursday is about local, not national, issues, but with Britain’s electorate in a volatile mood, the results will be watched closely. Were the Conservative Party of Prime Minister Theresa May to lose control of bastions like Westminster or the borough of Wandsworth, it could embolden those opposed to leaving the European Union and renew speculation about a challenge to Mrs. May, whose leadership has been questioned since she lost her parliamentary majority in June.