British Prime Minister Theresa May’s failure to get U.K. parliamentary approval for a deal to split the country from the European Union adds significant pressure on companies as they plan for the now greater possibility of an abrupt and disorderly exit.

Lawmakers on Tuesday soundly rejected Mrs. May’s agreement with Brussels and approved by her cabinet, putting the process in jeopardy ahead of a March 29 deadline to leave. The repudiation has opened up a variety of possible outcomes considered unlikely until recently, from...