Since Mountbatten is a straight translation from the German which was done in the 20th century, it's quite possible that the surname didn't exist in English-speaking countries before 1917, when, in order to get rid of their German-sounding names, the royal family and its relations anglicised their names and in some cases (the princesses of Schleswig-Holstein) dropped them altogether (or at least stopped using them). Prince Louis of Battenberg became Louis Mountbatten (and the house of Saxe-Coburg-Gotha became the house of Windsor and the Duke of Teck became the Marquess of Cambridge).
Leopold Duke of Albany (in real life) was called Leopold George Duncan Albert, which is different from the string of names in the movie. He was born in 1853, and in 1882 he married Princess Helen of Waldeck (the sister of Queen Emma of the Netherlands). He died in 1884. So I suppose he'd have been adult and single in 1876. However, he wasn't a Battenberg, so the Mountbatten surname is a bit incongruous. He was a Saxe-Coburg-Gotha prince, as the son of Prince Albert, and so his anglicised name would have been Leopold Windsor. But there's Hollywood for you...
