The interesting thing is that Britishness is really a Scottish (and yeah also NORN ION) concept; there was never a question of two identities in England or even Wales, where 'British'* just = as you were plus an extra flag with a bit of blue on it.
*As a national-political identity. Obviously all parts of the islands are part of a distinctive cultural region, whether part of the United Kingdom now or in the future or not. And had been so long before the Act of Union.
Since Cromwell imo, if not the flight of the Earls.
EDIT: At least for Ireland, perhaps have to wait for the highland clearances for Scotland? But this is something I know nearly nothing about...