I have heard people in Great Britain use the term 'British Isles' but it's not a phrase any Irish person would use, I guess because of not identifying as British. It's a tricky one. I guess the acceptability depends on whose point of view you consider. I know British papers will routinely describe people like Saoirse Ronan or Sinead O'Connor as British and there is usually an uproar about it here in the republic. I also find it confusing because Ireland is officially 'Ireland' legally, not the republic of, and then we have Northern Ireland, as a separate entity.
As an aside, I always love Moriarty drinking from that UK/ Ireland teacup precisely because of the complicated relationship between the two places and the prevalence of hidden enemies, informants and back-stabbings throughout the history of it.
No bother at all, I just thought I should say. American television is appalling at this, I still haven't gotten over the Gilmore Girls and the train they took from London to Dublin.