LONDON—The U.K.’s main opposition Labour Party is pushing plans to extend the state’s reach into the economy on a scale not seen in decades, as the country likely heads for its third nationwide election in four years this fall. The party wants to rewrite a rulebook that it believes created the economic hardship that fueled the Brexit vote.

Labour lawmakers and activists gathered this weekend for a get-together until Wednesday to discuss policy and settle on a new election strategy. The party has previously pledged to renationalize railways, water companies and the electricity grid.

It has called on the U.K. Treasury to steer private investment into favored industries. It wants workers on company boards and higher taxes on the rich. It has toyed with moving the Bank of England from London to Birmingham.

Behind the party’s shift to the radical left is John McDonnell, 68 years old and a close ally of party leader Jeremy Corbyn. Mr. McDonnell wants to become the U.K.’s first avowedly socialist finance minister in half a century.

Six years ago, he was a nobody in Labour, politically sidelined, recovering from a heart attack and preparing for retirement.