The British pound fell against the U.S. dollar Tuesday on mounting concern that the U.K. may break its ties with the European Union at the end of next year without a fresh trading agreement in place, clouding the nation’s economic prospects.

Sterling slipped 1.4%, its steepest daily decline in more than a year, to trade at $1.3153. The U.K.’s FTSE 250 index, which tracks companies with significant local operations, fell 1.4%—its biggest drop since Oct. 3.

Prime...