BRUSSELS—The European Union says it is ready for Britain to leave the bloc without a deal to smooth the split. Growing evidence says otherwise.

Until recently, few European officials believed a no-deal Brexit would happen, as their governments set in motion preparations. Now, Prime Minister Boris Johnson’s rising threats to take Britain out of the EU with or without a deal by the Oct. 31 deadline is prompting EU officials to dust off plans ranging from information campaigns to fast hirings of customs officers.

European...