Back when Britain voted to leave, “no one was talking about no deal as an outcome,” said Asa Bennett, who covered the referendum for The Telegraph, the daily British newspaper. “The consensus was that of course we’d have a deal. It would be patently obvious. It didn’t even come on the radar.”

The notion that a no-deal exit was an option was powerfully normalized by Mrs. May herself, as negotiations with Brussels bogged down, and the complexities of removal became more obvious. In a tough-talking January 2017 speech at Lancaster House, she adopted a warning that had been used only by a small group of hard-line euroskeptic campaigners. She warned that Europe should not try to corner Britain into accepting a “punitive” deal, because Britain would walk away. It represented Britain’s main point of leverage, since Europe would suffer severely from a chaotic separation.

“While I am confident that this scenario need never arise — while I am sure a positive agreement can be reached — I am equally clear that no deal for Britain is better than a bad deal for Britain,” she said. She followed this up by articulating a series of “red lines,” promising to exit the customs union, single market and European Court of Justice.

Mr. Redwood, an anti-Europe voice in Parliament for decades, said the Lancaster House speech marked the moment when his group’s thinking became fully mainstream.

“That was the point of the slogan that we got her to adopt,” Mr. Redwood said. “It was very important, and it wasn’t one speech; she said it consistently.” He added, “In those days, she was advised by Nick Timothy, who was very much on our side.”

Ms. Stuart, the Vote Leave leader, said that phrase, which was later included in the Conservative Party manifesto, set off a significant shift in public perception. “It did mean suddenly that talk of no deal was no longer seen as an extreme position,” she said.