LIVERPOOL—Just over two months into his tenure as U.K. prime minister, Andy Burnham is laying the groundwork for an unabashedly socialist overhaul of the British state. This week, a tieless Burnham strode across the stage at his Labour Party’s annual conference in Liverpool, delivering an emotional and upbeat pitch that eschewed the oft-repeated promises of short-term economic growth and instead focused on a long-term vision of big state intervention aimed at uniting Britain’s political left wing. The plan—which could fundamentally alter the country’s political trajectory—raises the prospect of years of tax increases to come and the U.K. state growing to…