LONDON—Paul Evans, a cardigan-clad clerk at Britain’s Parliament, spent most of the past four decades diligently studying arcane rules and conventions in the shadows of Westminster Palace.

These days, in the crucible of the U.K.’s Brexit crisis, the 63-year-old expert in parliamentary procedure has found himself one of the hottest commodities in British politics. “Now the butcher, baker and candlestick maker are suddenly interested in what I do,” he says.

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