I find it rather ironic that Trinidad and Tobago is a Republic that still defers to a British institution as it’s highest court of appeal, wheras Australia, which still recognises the Queen, is otherwise completely sovereign and defers nothing to Britain....
Fiji, at the moment, is a prime example where this might have been useful. It's current military government is relying on the traditional powers of the Royal Prerogative for legitimacy. Fiji no longer acknowledges the Privy Council - if it did, the Privy Council could rule that it isn't legitimate.
Those nations such as Canada and Australia which have developed their own robust legal traditions, eventually decided they no longer needed that particular protection. That doesn't mean they wanted to break with the Crown, however.