HONG KONG—The British pound tumbled dramatically in chaotic trading that included a flash drop in early Asian hours and sustained falls in London. Late morning in Britain, it was down 3% against the U.S. dollar.

The sharp drop in Asia came amid worries about the U.K.’s exit from the European Union that were accelerated by computerized trades, market watchers said. The pound fell more than 6% just after 7 a.m. Hong Kong time on Friday to as low as $1.1819 from just above $1.26, before recovering above $1.24, according to Thomson...