(written from a Production point of view)
The United States of Africa was the country on Earth in the 23rd century that was stated in Star Trek: The Original Series's writers' bible as being the birthplace of Uhura.  The first edition of the Star Trek Concordance, by Bjo Trimble, states that the Star Trek: The Animated Series episode "The Counter-Clock Incident" may have at one point referenced the United States of Africa, but the reference was apparently dropped from the final edit.
StarTrek.com, the official website, has also adopted the United States of Africa as Uhura's place of birth. This fact was also mentioned in the first edition of the Star Trek Chronology, by Michael and Denise Okuda.
In the 1977 Writers'/Directors' Guide for Star Trek: Phase II – the abortive second series – Uhura was noted as having instead been born in the "African Confederacy". This may have led to the naming of Geordi La Forge's birthplace as the African Confederation in Star Trek: The Next Generation episode TNG: "Cause and Effect".
In the real world, a formation of a United States of Africa has been proposed among the 53 African countries; forerunners of such a proposal are the Organization of African Unity (1963-2002) and its successor the African Union (2002 to the present).