TALLINN, Estonia—It took a £1 payment to uncover one of the world’s biggest money-laundering scandals.

Howard Wilkinson, a British trader at a Danish bank’s branch in Estonia, noticed that a London business, which moved more than $1 million through the branch almost daily, had filed a report with the U.K. government claiming it had no income or assets. Downloading the report cost Mr. Wilkinson one pound.

The...