Other answers have addressed the question pretty well in a general context, so I thought I'd give an answer focussing more on the linguistic history in particular
In this context, the diacritic is not referred to as an umlaut, but rather a diaresis (if indicating that the letter should be pronounced separately, rather than as part of a digraph) or a trema, whilst umlaut refers to a set of partially-grammaticalised partially-lexical vowel changes in the Germanic languages, some of which are shown with a trema in German orthography
There are three types of umlaut, i-umlaut, a-umlaut, and u-umlaut. The first two occurred in all surviving Germanic languages, whilst the latter only occurred in the North Germanic (Scandinavian) languages
I-umlaut affected vowels followed by an i (or a j, which in most Germanic languages has the sound of an English y). Back vowels (a, o, & u) affected by it move forward in the mouth (creating new vowels ä, ö, ü, in English these later became e, e, & i), whilst front vowels (e & i) move up in the mouth if possible (merging into i)
A-umlaut affected vowels followed by an a, and only affects the high vowels (i & u) which were lowered to merge into the mid vowels (e & o)
U-umlaut affected vowels followed by a u (or a w, which later became a v in most Germanic languages). Unrounded vowels (i, e, a) affected by u-umlaut became rounded (ü, ö, å)
In all cases, long vowels behaved pretty much the same as their short equivalents
If this was just easily predictable though, it wouldn't have ever needed to be written explicitly, you'd just know to pronounce an a followed by an i further forwards in your mouth. The reason it came to be written explicitly is because these vowel changes persisted after the vowels that triggered them were lost
As an example, the English word man has the plural men, reflecting an earlier stage where it was mann and manniz with the i triggering i-umlaut. The -iz ending was lost, but the umlaut from a > ä (which later developed into e) was retained
German reduced unstressed i to e in most cases, so most instances of i-umlaut that still had an obvious trigger seemed to have it triggered by an e, and so German scribes started writing an e above a vowel to show that it had been affected by i-umlaut and this eventually evolved into the modern German trema we see today
The diaresis as used in French (to show that the second vowel letter in a sequence should be read separately, rather than as a digraph) is a direct continuation (via Latin) of the Greek trema used in the same way developed during the Hellenistic era (along with most of the rest of the Greek diacritics)