OK, here’s what I think is the real answer. In English, the umlaut is used to indicate a diphthong in pronunciation of adjacent vowels. Hence, ‘meat’ does not take an umlaut because ‘ea’ is pronounced as one sound: roughly, ‘eeee.’ By contrast, when you say ‘naïve’ the ‘ai’ actually makes two vowel sounds—roughly, ‘ah’ and ‘ee’—so you put an umlaut over the ‘i’ to indicate the transition to a new vowel sound. Otherwise it’d be pronounced ‘nave’ or ‘nive’. Similarly, you put an umlaut over the second ‘o’ in ‘coöperate’ to indicate the transition, between the two O’s, from the ‘oh’ sound to the ‘ah’ sound. Otherwise it would be pronounced ‘koop-er-ate’. Like a chicken coop. Of course the umlaut has in large part been dropped from popular spellings of these words, but many organizations retain it (e.g., the New Yorker magazine).
So to answer the question: there is no umlaut on ‘umlaut’ because the ‘au’ makes roughly one vowel sound, rather than requiring a hard transition from one sound to another between the two vowels. The one sound being like the ‘o’ in ‘ow’. As to andrew’s suggestion that the first U might take an umlaut, I can’t think of any reason why that would occur in English, except if a given word were a direct, unaltered, and recent transplant from another language (most likely, as andrew rightly notes, German or Hungarian).