As a native english speaker, Å. Ä, and Ö don't have much meaning for me, other than the fact that they are regular A and O's with funny dots over them. When necessasry, I pay attention to them, but most of the time, I just ignore them. It really doesn't make much difference to the word I hear/see in my head.
For example, today the bus crossed the street: "Båtmansvägen."
I read, "Batmansvägen," and couldn't help but look around for the Batmobile.