Also, as opposed to German, they are not referred to as umlauts or accents. ä is a completely different letter compared to a, and they have absolutely no relation. An indication of this is that I can't even think of a word that describes the dots over the a, similar to how most English speakers wouldn't be be able to name the dot over the i.
As such, a search or sorting algorithm that puts equivalence between ä and a would be completely broken. However, an English speaker who only wants to look up a Swedish word in a dictionary would definitely want that equivalence. They may not even be able to type the Swedish letters. From this I conclude that it's the locale of the user that must dictate the way comparison works, not the language of the words that are being compered. This is thankfully how locales typically work.
In Swedish, these are extremely common letters, there are plenty of examples where words differ only in letters which would change using such algorithms. Such as älg/alg (elk/algae), or kö/ko (queue/cow).