Firstly it's worth noting that Swedish doesn't have umlauts like that and it's a separate letter completely. But disregarding the semantics we have officially used Ä and Ö instead of Æ and Ø since at the very least 1541.

Which is when Gustav Vasa's Bible was written that set new standards for the Swedish language, and the change to Ä and Ö was partly motivated by seeking a stronger separate identity from the Kalmar union (which was a Union between Norway Sweden and Denmark that we had just fought ourselves free from due to the Danes effectively controlling it).