Maybe we could have one African city state emerge around 1500-1600 and rapidly develop through trade with Europe, while investing most of their profits into infrastructure and literacy. Play Europeans off against each other, implement new agriculture, slowly expand between 1600-1800. Have a centralized state with several hundred thousand miles worth of land under their control, 5% literacy, ten million or so people, the ability to produce small arms in small quantities, and a few major trade routes.
From 1800-1850 they more than double their land through conquests and have a million square miles by 1850 and 30 million people. Give them the ability to mobilize several hundred thousand men with guns, even if outdated. That in unison with disease makes them very difficult to actually conquer. They continue focusing on long term investments and allow trade with Europeans.
From 1850-1900 they conquer another million square miles and control over 2 million square miles. 60 million+ people by 1900, 10% literacy, several major rail roads built, and a handful of factories.
From 1900-1950 they join two world war equivalents, going on the winning side, and seize another million square miles of land to control around 30% of the continent. Literacy rises to around 20% while the population is around 120 million with a GDP per capita of $1200 current USD. The 50s and 60s end up very prosperous with huge increases in income levels, literacy, and life expectancy. By 1970 this country had a GDP per capita of around $2500 current USD, 50% literacy, and a life expectancy of 60.
Independence movements idolize this hypothetical country. In the late 50s it’s leaders for whatever reason create an open border free trade union with a defensive military alliance for anyone who joins. It requires just a few rules about industries and assets that cannot be nationalized, agreements to protect property.
In the 60s newly independent countries begin joining left and right. Countries that do tend to see upticks in growth, while those that don’t tend to do poorly. 80% of Africa has pledged to join by 1990, with a majority of it in. Huge gaps emerge between early members and those who haven’t joined by that point. More standards for long term members, like offering universal grade school education, are enacted. By 2000 the last of the continent joins.
A shared budget and taxing power is implemented in 2000. In 2005, schooling is made compulsory and paid for by the African Union from the K-6 level. Rail and highway connecting every part of the continent are complete in the early 2000s.
Due to issues maintaining stability, protecting property, lives, and trade routes, and responding to disasters, a unified military is decreed in 2003. By 2008, it has ensured order everywhere on the continent. By 2015, it continues to grow beyond the original idea and surpasses all national militaries combined, with many national militaries being disbanded altogether.
By 2015, parts of the continent lose their Developing status and reach High Income. Most of the worst poverty is gone by 2015, and no longer covered by foreign aid. A common currency is established in 2016. Nationalization of private industry has to be approved by the national government and compensated as of 2016.
This wildly unrealistic scenario might be the best we can do. Not technically one country, but probably the closest Africa can get given its size, disease, and isolation.