LONDON—British lawmakers voted Saturday to postpone a decisive Brexit vote, likely forcing Prime Minister Boris Johnson to request a further delay of the U.K.’s departure from the European Union. Parliament said it needed more time to review a deal Mr. Johnson concluded this week with European leaders that sets out citizens’ rights, a financial settlement with the EU and a special arrangement for Northern Ireland that would require customs checks on goods arriving there from elsewhere in the U.K. The move was backed by opposition lawmakers, along with some who have recently left or been expelled from the ruling Conservative...