South Africa became independent years before (1910?) and Ghana had gone in '57. Apart from the Maumau uprising in Kenya most UK African colonies were not unduly restive and seemed to be accepting of the UK policy of 'as and when ready'. After all Britain had shown itself willing and able to release countries before viz. Australia, New Zealand, Canada and India/ Pakistan. Up to this point the UK had avoided the 'domino effect' and independence for Nigeria in 1960 was seen as logical and not before time. The precipitating factor for the further changes between 1960 and '63 was not so much African opinion as international opinion. There had been nationalist movements in other parts of the world and 'colonialism' had become a dirty word along with 'paternalist'. The fact that the removal of British rule led to inter-tribal bloodbaths and rampant racketeering by gangster governments was not seen to invalidate (even in hindsight) the decision to start a process of general withdrawal.
The biggest single trigger was the decision by France, locked as it was in the Algerian crisis, to pull out of Africa if the Africans were willing to accept independence. Almost the entire French political presence in Africa ended in and during 1960. (Culturally, linguistically, and economically it never left, of course.) (This is Dot's point - suddenly Britain was very obviously the major colonial power in Africa, even if it was largely tolerated within Africa.) This was the background to a series of independence ceremonies over the next three years in Sierra Leone, Tanganyika, Uganda, Kenya and Zanzibar. 1963 is rather an arbitrary end-point. Malawi and Zambia didn't go until 1964, Gambia in 1965 and Botswana amd Lesotho in 1966. From the point of view of the administrative burden the turnover rate was pretty rapid, but nowhere near as precipitous as France's and not really in keeping with a word like 'surrender'.