The Confederate States targeted the Bahamas early in 1941, at the beginning of the Second Great War. By the end of the year, the Bahamas had fallen to the C.S.. This handicapped the USA's ability to operate in the Atlantic generally and the Caribbean specifically for the first years of the war. Native Bahamians, who were generally black, continued to resist the CS presence.
It is not entirely clear who rules the Bahamas in the Worldwar timeline after 1944. At the time of the Race Invasion of Tosev 3, it was under the rule of the United Kingdom. After the Peace of Cairo, the U.K. lost a number of its territorial possessions. In Second Contact, only three independent countries in the Caribbean are listed: Cuba, the Dominican Republic, and Haiti. It is also stated that what the Race didn't occupy was taken by the Germans. However, the Race was generally not equipped to occupy islands in the Worldwar series, and nothing suggests Germany colonized the Caribbean. The map included in Second Contact is not helpful, as the Bahamas are too small. However, it is entirely possible that the Bahamas remain under the rule of the United Kingdom after the Peace of Cairo, since the United Kingdom didn't join in the Race-German War of 1965.