Why did they change their family name from a German name?
House of Saxe-Coburg and Gotha (/saːksˈkoʊˌbɜrɡəndˈɡoʊθə/; German: Haus Sachsen-Coburg und Gotha) is a German dynasty, the line of the Saxon House of Wettin that ruled the Ernestine duchies including the duchy of Saxe-Coburg and Gotha.
Founded by Ernest Anton, the sixth duke of Saxe-Coburg-Saalfeld, it is the royal house of several European monarchies, and branches currently reign in Belgium through the descendants of Leopold I, and in the Commonwealth realms through the descendants of Prince Albert. Due to anti-German sentiment in the United Kingdom during World War I, George V of the United Kingdom changed the name of his branch from Saxe-Coburg and Gotha to Windsor in 1917. The same happened in Belgium where it was changed to "van België" (Dutch) or "de Belgique" (French).
Where's the German ancestry in these children on their mother's side?
Even on their father's side their great-grandfather was born on the Greek island of Corfu and their great-grandmother's mum - the Queen Mother - was more Scottish than anything.
There is no way these children are German. It's plain to see in black and white.