Wikipedia states "The Oxford English Dictionary (OED) states that the earliest recorded sense of the word 'English' is "Of or belonging to the group of Teutonic peoples collectively known as the Angelcynn [...] comprising the Angles, Saxons, and Jutes, who settled in Britain during the 5th c.". However, the OED continues that...."
So anybody claiming to English is also admitting to be of immigrant decent.
My surname came over with the Normans in 1066 - One of Wm the Conqueror's generals shares the same surname as me, so I am of immigrant stock, too!
Just as "France" is a corruption of "Franks" a tribe that lived in Germany!
Edit : For those that claim to be "pure" Scottish, ain't no such animal, the Vikings pretty much settled all of Scotland and the Scottish surname "Wallace" is a literal translation of "Welsh" Or Irish descent, like Graham Norton, perhaps? "Irish folk music" has the same roots as Spanish Flamenco music - brought over by "gypsies" who originated in India.
Red-headed celts? No, Viking descent.
Robert the Bruce? Scottish? No, a descendant of William the Conqueror.