(written from a Production point of view)
The United States of Africa was stated in the original series' writers' bible as being the birthplace of Uhura, but this has never been confirmed by any on-screen reference. This would also be picked up on in the first edition of the Star Trek Chronology, by Michael and Denise Okuda. StarTrek.com, the official website, has also adopted the United States of Africa as Uhura's place of birth. Regardless, it has not been expressly stated in canon.
The first edition of the Star Trek Concordance, by Bjo Trimble, states that TAS: "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.
It is interesting to note that 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.
A formation of a United States of Africa has been proposed among the 53 African countries; forerunners of a such a proposal are the Organization of African Unity (1963-2002) and its successor the African Union (2002 to the present).