LONDON—The long-cherished goal of successive British governments to reduce the nation’s decadeslong habit of spending more than it earns may be coming within reach, but not necessarily in the way they would have liked.

The U.K.’s decision to leave the European Union could make it harder for the country to sustain its large and persistent current-account deficit, which means it brings in less from overseas trade, investment income and remittances each year than it shells out to foreigners.

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