OK, so let's say that for the purposes of this question, we are dealing with a hypothetical TL where in addition to the UK and the three OTL Crown Dependencies, there would also be Puerto Rico as a British Crown Dependency. How that comes about is not relevant to the question.
Now, in OTL we already have the "British Islands" to refer to the UK + Crown Dependencies. Would it be any different with Puerto Rico as a British Crown Dependency?