Prior to the adoption of Mountbatten in February 1947, Prince Philip did not use a surname. In the navy during World War II, he was known as HRH Prince Philip. Thus, while serving as First Lieutenant on HMS Wallace (Oct. 1942 to Jan. 1944), he was Lt HRH Prince Philip, RN. He was also referred to as Sub-Lieutenant Prince Philip of Greece in the 12th of August, 1942 edition of the Dundee Evening Telegraph.
Nonetheless, some sources misleadingly refer to him as Philip Mountbatten while he was serving on various ships during World War II, despite the well-established fact that this name was not adopted until 2 years after the war had ended.
The non-use of a surname dated back to his childhood. At the age of six, he was sent to a "a progressive American kindergarten" in Paris called Elms, where
Asked to introduce himself in class he insisted at first that he was
‘just Philip’, before eventually awkwardly admitting that he was
‘Philip of Greece’.
Source: Philip Eade, 'Young Prince Philip: His Turbulent Early Life' (2011)
Later, at Salem’s junior school, Spetzgart,
Philip was known by his first name by the English and American boys,
whereas the Germans addressed him simply as ‘Greece’.
while at his next school, Gordonstoun, he
was listed simply by his Christian name on the school register
In his private life, he was also known simply as 'Philip' or 'Philip of Greece' (according to the Canadian debutante Osla Benning, with whom Philip was romantically linked).
On the adoption of the name 'Mountbatten',
Philip might have been expected to take his paternal dynastic name of
Schleswig-Holstein-Sonderburg-Glücksburg. This, though, would have
looked incongruous on a British passport, and by drawing attention to
his German ancestry with the trials at Nuremberg only recently
concluded it would have been distinctly unhelpful to his cause.
Genealogists at the College of Arms had come up with the snappier
alternative of Oldcastle, an anglicized version of Oldenburg, the
German duchy where the royal house of Greece and Denmark had
originated, but this did not find favour either; as much as anything
because it was thought to sound slightly plebeian.
Eventually, the name Mountbatten was settled on, this being the anglicized version of his mother's name. Even so,
Philip, meanwhile, by his own account, ‘wasn’t madly in favour [of
taking the name Mountbatten] … but in the end I was persuaded, and
anyway I couldn’t think of a reasonable alternative’.