LONDON—The fall of crude prices to six-year lows in recent weeks has pummeled oil-producing countries from Venezuela to Iraq, threatening their ability to keep pumping petroleum and forcing governments to take painful measures to shore up national budgets.

Few countries have been as spun around as Nigeria, Africa’s top oil exporter. The nation of 182 million people depends on oil for 75% of its revenue, money it needs to roll out new infrastructure for a population that grows by 13,000 people a day. In April of last year,...