2400 year ago Britain was inhabited by the Celtic tribes who came across from Europe in 300 BC. So the first Britons were European.
The Angles and Saxons came across from Germany, Netherlands and Denmark to displace the Celts.
Then the Romans came across from Italy, the Vikings (mainly in the north) from Norway, Denmark and Sweden and the Normans from France.
Then there were the major mass immigrations – the Huguenots from France and Jews from Russia and Eastern Europe.
The trade links of the British Empire opened up trade routes and there was an influx of people from further afield – Arabs, Indians, Africans, Chinese and hosts of others.
In the 20th century we had further waves of immigration from Jamaica, Pakistan, India, Africa, Turkey, Greece and the Middle East.
The stand out thing about the British is that we are mongrels, full of hybrid vigour. All these people have contributed greatly to our culture. Our language is the best example. It is basically German/Danish but with much French, Italian, Indian, Jewish, Arabic and Dutch imports. This provides the richness of language that enables nuance and an array of subtleties that most other languages do not have. It is this richness that gives us writers like Shakespeare.
To be British is truly to be European in every sense of the word.