I am thinking they might side with Denmark during the Scleswig- Holstein wars 1848-51 and 1864. If the Swedes and Norwegians help Denmark keep the duchies (or perhaps just Schleswig would be enough) there could be some butterfly effects creating a Scandinavian Union centered around Sweden. Best case scenario the wars goes even better and the Danes keep Holstein and Sweden reconquer their Pommeranian provinces. I'd say a kingdom consisting of Sweden, Norway, Denmark, Schleswig-Holstein, Vorpommern and the Scandinavian Atlantic Islands would be a good beginning for a power in the European Concert.
Let's say Sweden has been even more involved in the conflict, not just fought the Prussians, but also participated in the Crimean War 1853-56. Let's say that Finland was reconquered with Brittish and French help. If you then add finland to the equation earlier you have a great power. Then again this Scandinavian nation would have created dangerous enemies in both Prussia and Russia. Perhaps it would not last too long, unledd it's allied with the UK of course. Then the Scandinavian Union could even be a Entente Power in WW1 (if Germany is unified, might not be).