Being British seems to be an idea that all who live in Wales, Northern Ireland, Scotland and England, Cornwall too, can call themselves British if they subscribe to the Union of these countries under our Monarch, Her Majesty the Queen. Of course there are Welsh nationalists, English nationalists, Cornish nationalists, Scottish nationalists, an perhaps even Northern Irish nationalists or those who seek nationalism with Southern Ireland who may chose not to call themselves British at all. So those who wish to be part of the United Kingdom are all British.
Much of the regional sections like Cockneys, Brummies, Geordies, etc of the 1970s that I came and saw in the United Kingdom have long vanished, and even Scots do no longer have their Engish dialect so I do not know what present day Scottish National Party represents culturally when the entire United Kingdom has been homogenised. We Indians have also lost our accent and assimilated into the British way of life.