Few of the British colonies took the name “Republic of..”, most took the name “Commonwealth of..” as in “Commonwealth of Australia.” A few took “Dominion of...”, as in Canada. “Republic” implies a rejection of the Crown, which the UK wanted to avoid, and so prevailed upon newly independent nations from her empire to avoid such terminology.
Actually, in Africa, most (if not all) of the former British colonies became republics and used that word in their offical names. That includes the Republic of Botswana, the Republic of the Gambia, the Republic of Mauritius, the Federal Republic of Nigeria, the Republic of Seychelles, the Republic of South Africa, the Republic of the Sudan, the United Republic of Tanzania, the Republic of Zambia, and the Republic of Zimbabwe.