Generally, you need to distinguish between the sounds found in the umlauts themselves, and the Germanic process of umlaut. The former are obviously cross-linguistically more widespread. For example French, Hungarian, Turkish, and it should also be pointed out that (Hellenistic/Medieval) Greek must have possessed the sound for the letter Upsilon (Υ), wheras in modern Greek, the letter is pronounced like an /i/ (the convention to pronounce "Y" as "ü" was retained in German, if you look at words like "Hypnose" or "Apokalypse").
Now, the process of umlaut (ie, in German represented by a > ä, o > ö, u > ü) is a uniquely Germanic feature, and it's actually a fairly late development that was clearly not present in Proto-Germanic: the Gothic language (attested from the 4th century AD) lacked the umlaut feature, but other, later Germanic languages have it. It's obviously visible in German:
Schlag > Schläge
Loch > Löcher
Kuh > Kühe
Even English retains (though considerably modified) the process of the Germanic umlaut, for example:
mouse > mice
louse > lice