Why is the letter W pronounced "double-u" instead of "double-v"?

It has to do with its function, not its appearance. From The Origins and Development of the English Language, by John Algeo:

The history of the curved and angular forms of u -- that is, u and v -- was similar to that of i and j. Latin consonantal and vocalic u came to represent quite different sounds early in the Christian era, when consonantal u, hitherto pronounced [w], became [v]. Nevertheless, the two forms u and v continued to be used more or less interchangeably for either vowel or consonant. As its name indicates, w was originally a double u, although it was the angular shape v that was actually doubled, a shape we now regard as a different letter.