It creates an interesting scenario.
Russia would probably side with Prussia - they are smarting from the lack of Austrian support during the Crimean War, with them having helped put down the Galizian and Hungarian revolts in 1848 - besides, Russia does not want to see either Austria nor France grow too powerful.
Italy was a Prussian ally against Austria 1866, but they will hardly go against France, which holds substantial land and troops inside Italy at this time. Neutrality is probably their best option.
The Ottomans are reforming after the Crimean War and are probably not after another conflict at this time. They most likely stay out.
The big wildcard is Britain, which controls about 50% of the total industrial production of the WORLD at this time. If Prussia fights alone, they might side with Prussia to ensure the balance of power. But a Prussian-Russian alliance might have them side with Austria-France-Denmark, as they have cordial relations with France and still view Russia with suspicion since the Crimean War.