Well, first off a surname is only indicative of a small portion of your ancestry that has been passed down through the generations via tradition and marriage (i.e. your father's father's father and so on and so forth), so if there has been a great deal of intermarriage down through the generations, the original ethnicity that the surname is derived from may only represent a minor fraction of one's DNA.

However, the much more likely explanation in your case is that your English/British ethnicity is showing up as other DNA from adjacent geographic origins, such as Scandinavia, Ireland, or Europe West. The way I've come to interpret what "Great Britain" DNA means is that it's indicative of the Celtic-speaking populations who lived in Great Britain prior to the island's conquest by the Romans and then later on by the Germanic Anglo-Saxons, Danes, and Normans. These groups were known as Britons, Picts, and other tribes. Maybe your ancestors included some Irish settlers, Germanic Anglo-Saxons, Vikings/Danes, Norman-French (themselves a Scandinavian Viking/French blend), and others who settled in Great Britain later on and hence would show up under that DNA.

Therefore, there's no reason for you to assume that the English ancestry isn't there, but simply that it's showing up as something else. For example, in my case I also have some British ancestry that I suspect is partially showing up as Scandinavia DNA (I have no recent Scandinavian ancestry). Likewise, I also have a lot of Italian ancestry from Sicily and to a lesser extent the north which is partially showing up as Caucasus/Middle East and also probably Iberian Peninsula (two other regions where none of my immediate ancestry comes from but is probably a legacy of such events as ancient Neolithic migrations, Ancient/Byzantine Greek colonization, Phoenicians, Arabic Moors, Spanish rule in Sicily, etc.).