Well, if we go for a post-1900 although pre-independence POD, if French West Africa (
http://en.wikipedia.org/wiki/French_West_Africa ) was governed in a more centralized fashion and local "evolvees" were sent to other parts of the state or accumulated in the capital to create a more homogenous elite, it might have emerged from the colonial period as a unified state: it'd be no more ridiculous, really, a conglomerate than Nigeria or Zaire.
With over 100 million inhabitants OTL, and given that some of its components (Senegal, Ivory Coast) actually have a higher GNP/Capita than Pakistan, I don't think we need a leadership of
absurdly high competence (by African standards) to get a nation with the human and material resources to build a bomb (especially if the Pakistanis/and/or N. Koreans are a buyers market for nuclear info): they have the uranium in OTL Niger, after all.
Tricker is motivation: national prestige, sure, but it's
pricey for third-world regimes - no Latin American state ever built one. Perhaps a fierce African leadership contest vs. the Nigerians? (Into which we can add an AngloSphere vs Francophonie contest

). Perhaps a longer-lasting Apartheid regime which develops its own bomb, thereby providing incentive for the "African bomb?"
Now, POST-independence, its rather harder: no African unions or federations have lasted OTL. (BTW, have you ever read Mack Reynold's 'Black Man's Burden' series?)
Bruce