It induces fronting.
/ u / is rounded, high and back in the mouth. The IPA / y / is rounded, high, but front in the mouth. French < ou > v < u >.
Same for / o /, rounded, back, and mid versus the umlauted variety: both are rounded and mid, but the umlauted variety is fronted. French also has both phones.
/ a / is a bit different because the low vowel space is more constrained. You'd think the fronted or 'umlauted' version would be /ae/ (as a single glyph) but instead it's raised in most dialects into the phonetic space occupied historically by some mid-front unrounded vowel. Maybe it resulted in a (likely) push chain that caused speakers to merge the mid-front unrounded vowels to make space, maybe it merged with one of the unrounded front vowels.
Usually the German umlaut resulted from compensatory lengthening, so that the resulting vowel was lengthened and fronted. (We had lengthening in English from loss of final < e > (for want of a better representation), and in some cases it did affect the vowel quality and not just vowel length. In some cases the change was levelled, in some classes cut-syllable timing did the trick or tri-syllabic shortening overrode the outcome. But it's been a long time since I looked at English historical phonology; my German h.r. is much more recent.
How rounded front vowels are borrowed into other languages or how they change over time within a language varies in ways I can't explain and haven't always seen explained satisfactorily. Take "revue", with its / y / (same as umlauted < u >
. In English it comes out as "you"; but in Czech it's "ee". "Revue" might be "ree-'vyue" in English, but it's clearly "'reh-vee" in Czech.
(But I'm a physic sciences high school teacher, so what do I know?)
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