Metal umlauts!
I answer this as someone who studies both Metal and Germanic cultures.
So what is the actual usage of the umlaut? Well the umlaut is used in most notably in the German written language as well as other Germanic based languages such as Swedish, Icelandic and Faroese. (Finnish does use umlauts too but it is not a Germanic language) There are other uses for the umlaut in other languages such as Turkish but those won't be useful in our analysis of it's use in metal.
So the first instance of the use of umlauts in a metal group began with the name of the Blue Öyster Cult whom the manager Sandy Pearlman at the time of coming up with the name was noted saying "How about an umlaut over the O?' Metal had a Wagnerian aspect anyway." Another instance is that of the name for the metal band Motörhead whom Lemmy from a 2002 Wave interview said "I only put it in there to look mean." Another famous use is with the name Queensrÿche which invokes a misspelled version the German word reich (realm or kingdom in English).
So how do we compile this information? Much of early "metal culture" is heavily based in countries where cultures with Indo-European languages are common place and many of the musicians had cultural ties to that of Western Europe. (America being tied to Europe through mainly England and England being tied through their "Anglo-Saxon" identity tied to the area of Germany/Denmark) As such there is a fascination in metal with the "barbarian" and "teutonic" elements that came from musicians Germanic heritage, imagined or not. This fascination reaches to idealizations of the brutality invoked by the Goths, Vandals, and Huns of the Migration period or the Vikings (Norse) of the early Medieval period, etc. The use of umlauts is the metal musicians way of tying the heaviness of their music to invoke the notion of Furor Teutonicus. If the tie is purposeful varies as the "mean-ness" of the umlauts use eventually perpetuated its own intrinsic meaning in the metal scene to where the use of the umlaut is not even understood for it's actual use. The anecdote from Lemmy backs up this notion. My favorite story of the ignorant use of metal umlauts is the from the interview in Vanity Fair with Vince Neil in 2009 where he talks about going to Germany with Mötley Crüe where they were confused by fans screaming "Mutley Cruh! Mutley Cruh!" which is the correct German way of saying what they spelled.
tl;dr Umlauts invoke a subconscious feeling of Furor Teutonicus in metal heads.