Both Nigeria and Kenya had seen limited forms of self-government before independence.
For example, Nigeria's first post-independence President, Nnamdi Azikiwe, had previously been a journalist and publisher and then a politician and colonial Governor-General, while its first Prime Minister Abubakar Tafawa Balewa had previously been a teacher and school inspector and then a politician and government minister.
Meanwhile in Kenya, the first post-independence Prime Minister and a year later its first President, Jomo Kenyatta, had a university education at the London School of Economics and then became a politician, being imprisoned by the British during the Mau Mau uprising but after his release being elected to the Legislative Council.
So in both these cases, and in many similar cases across the British Empire, many of the leadership immediately post-independence had political and government experience before independence. The same was the case in much of the French Empire, and in both cases there were also some attempts to educate an indigenous professional class, notably in universities in London and Paris.