I gust had in idea that if sweden had supported Denmark (even if they lose) pan-scandavnisum would get a bost, but bast on the replys on this thread it appears to not be to likely.
From my understanding, there would be hurdles for anything major. Even Norway already had its own parliament and government by this point, only being in personal union with Sweden. It might be possible for a military alliance to form, but the main issue is the different geopolitical situations of Sweden and Denmark. This war all but showed Denmark that there was really no point in maintaining an effective army as there was really no way they could hold off the Germans, so 'what's the point?' Sweden maintained an armed neutrality, which might seem the same but Denmark is far more likely to be invaded than Sweden. A military alliance would mean that Sweden would then be compelled to come to Denmark's aid.
And as most have said, even all the Scandinavian countries together can't even really challenge even the Prussian coalition of this time, much less than later German Empire. The Scandinavian countries were still quite poor at this time, and even together I'm not sure they could even be a true secondary power on the level of say, Romania. Meanwhile all the Scandinavian countries learned during the Napoleonic Wars just what happens when minor powers allow themselves to be caught between Great Powers (all of them basically got screwed during the wars, and were all but disregarded by the Great Powers during the Congress of Vienna besides arranging things to satisfy the balance of power). So Sweden and Norway have little reason to raise their risk of being forced into a war by forming a military alliance with Denmark.
It might be possible, although I admit I'm probably not the best person to judge. The 2nd Schleswig War all but caused the adjustment policy of Denmark to appease Germany. If Sweden-Norway had sent aid, even if they lost, it might might have increased the perception among the Danes that if nothing else maintaining an effective military could deter any casual invasions or at least hold with Sweden-Norway long enough for foreign aid to come. However this is still predicated on the needed guarantee that Sweden-Norway would come to their aid in such an event, something difficult to achieve since Charles XV in the Schleswig War made promises to do so but ultimately saw the danger of the proposition.
Scandinavianism is often interesting during times like the Winter War or World War II where there is a perception that greater cooperation could have had an effect to their benefit, although the Winter War is more pan-Nordic, but really a more successful Scandinavianism movement would likely prevent such situations in the first place.
More likely than a military alliance is more economic integration. Norway and Sweden were bleeding people to the US during the second half of the 19th century. Greater sense of Scandinavianism alongside hopes of cooperation might encourage more to stay, and there might have been some possibility that more cooperation could lead to earlier industrialization. They already did pretty good in regards to language, so I'm not sure much would change there. The different languages are more for national pride, and are still similar enough for relative ease of communication. As for politically. Still doubt there'd be any real unification, and Norway is still probably going to break the personal union with Sweden. You might however make it a more agreeable split, since despite avoiding conflict there was still something of an diplomatic snafu between Norway and Sweden when they split. An earlier Nordic Council would probably be the best a pan-Scandinavia movement could achieve. Maybe later you could have them unify their foreign policy and defense after an alt-WWI, the US actually put an end to some minor discussions for this OTL on behalf of NATO, but its very unlikely that any of them would give up their sovereignty for a real union. A pan-Scandinavian parliament-forum for cooperation alongside a lifelong alliance seems pretty extensive already.
Although one potential political effect might be in regards to North Schleswig. In the treaty ending the war, Denmark had the Germans agree to a vote that would allow North Schleswig to vote whether it wanted to return to Denmark. Prussia and Austria just didn't do it, and so it wasn't till after WWI that the vote went through and it returned to Denmark. If Sweden-Norway sided with Denmark in presurring them to allow the vote, it could have potentially been returned to Denmark decades earlier.