LONDON—British Prime Minister Theresa May delayed a critical parliamentary vote on her proposal to leave the European Union, throwing both her government and her plans for the U.K.’s exit from the bloc into disarray.

The postponement came after several British cabinet members urged Ms. May to pull the vote scheduled for Tuesday rather than suffer a potentially large parliamentary loss that risked toppling her government and throwing Brexit talks into even greater chaos. The British pound dropped to its lowest level in more than a year on the news.

“If we went ahead and held the vote tomorrow, the deal would be rejected by a significant margin,” Mrs. May told jeering lawmakers in the House of Commons after she spent an hour with her cabinet.

With a March 29 deadline for the U.K. to quit the EU, a fresh round of fevered negotiations is now needed to adjust the deal and present it anew to Britain’s Parliament.

Mrs. May plans to meet with European leaders before an EU summit in Brussels on Thursday to try to drum up support for changes. She will seek assurances from them that a last-resort “backstop” guarantee that ensures no hard border appears between the Republic of Ireland, an EU member, and British-ruled Northern Ireland after Brexit will never be needed. This, she says, will persuade British lawmakers to back her deal.