KETTERING, England — Having emptied their bowls of raspberries and cream, the Conservative Party members who had gathered in the garden of this Elizabethan manor house turned to face the dance floor. The smell of sheep grazing nearby drifted into the tent. A pile of raffle winnings — pruning shears, sparkling rosé — waited beside the bandstand. Near the end of the local Conservative association’s 40th annual raspberry and wine night, the place went quiet as the chairman — man or woman, it’s always chairman — announced the winning tickets.

But jazz, drink and festoons of Union Jacks were poor covers for the discontent coursing through the party membership in Kettering, a town in central England that voted six-to-four in favor of Britain’s leaving the European Union. Gathered here were three dozen of the 160,000 or so party members who will choose the next Conservative leader and, therefore, prime minister, giving them unparalleled power to determine their country’s fate as it careens through the Brexit crisis.

This sliver of the population, just 0.3 percent of registered voters, is mostly white, aging and male. And it is poised to use its new clout — the party’s grass roots have never before picked a prime minister — to catapult Boris Johnson into Downing Street, potentially cleaving the world’s oldest and most successful political party as it sends Britain on the path to what could be a tumultuous Brexit.

As the Brexit mess has unfolded, Conservatives have grown ever more impatient, ever more fixated on leaving the European Union, come what may. Most members said in recent polling by YouGov that leaving the bloc was worth enduring significant damage to the economy, secession by Scotland and Northern Ireland and even a shattering of the Conservative Party itself.