LONDON — Prime Minister Theresa May of Britain survived a critical vote in Parliament on Tuesday that could have delayed Brexit, undermined her strategy for leaving the European Union and undercut the country’s constitutional protocol.

Mrs. May’s latest political escape came when lawmakers narrowly failed to approve an amendment giving Parliament the power to instruct her to seek a delay to avoid a disorderly, and possibly chaotic, exit that Britain faces on March 29 if there is no agreement.

However, she later lost a vote on a nonbinding amendment that said Britain should not leave the bloc without a deal, a sign of potential troubles ahead.