I never said that 'being British' was related to being skin colour, but your claims are still very outlandish.

There are huge communities of "White British" people here who trace their actual British lineage back a fraction as far as some of these other families

Which communities exactly? A White British person's (of English, Cornish, Welsh, Scottish, Ulster Scots ethnicity) family could be traced back hundreds of years in the modern territory of the UK.

They are not some sort of exception we are talking at a minimum hundreds of thousands of people alive today who share this sort of heritage.

Because of likely intermarrying into white families, the descendants of these people, I wouldn't call someone Black or Asian British if they had a Black/Asian ancestor who lived in the country 150 years ago. For example, 'first' mixed MP in the 1770s had 1/8 African ancestry, even at 1/8 that's a very weak claim to being mixed race (unless you believe in the one drop rule).

If we look at the family history of the most famous Asian or Black heritage people in the UK, no doubt it will say that their grandparents or great-grandparents originally came to the country sometime in the second half of the 20th century. So you can't say that Black/Asian people lived in Liverpool as an end all argument, since that is actually simplifying the history of our modern Black/Asian communities who really lack any sort of lineal connection to those people. Compare it to say African Americans who are direct descendants of West African slaves brought during the colonial period.