Russia: Korea was kind of on the far end of Russia's already sparsely held Far Eastern holdings. Its possible, certainly, but Russification would have been difficult to pull off. The Koreans are just too culturally solidified and centers of Russian influence too far away. I assume this would also involve the annexation of Manchuria (to make a land connection), which also runs the risk of such an annexation creating problems with Britian and France (Who see Russia even further disrupting the balance off power), leading to them either backing the Japanese or Chinese to check Russian expansion.
China: Considering how long China had dominance over the peninsula and didn't annex it, I doubt they'd do more than just leave Korea as a client state. The political motivations just don't seem to be there.
France: Seems like a likely candidate, as they would have a motivation to expand into the Sea of Japan/Central Pacific area for commercial and strategic reasons. If they push for greater influence in China, annexing Korea could be a good "springboard", or a sop offered by the British to compensate them for growing Anglo dominance or accepting the British-backed Emperor's victory over the pro-French Shogun in the Boston Wars (French-friendly Korea to counterbalance the British-friendly Japan). Considering how the French operated in Africa and Indochina, they seem the most likely to adopt a policy of Frenchification over their new subjects... though the results wouldn't likely be as impressive as hoped.
Britain: If Japan doesn't align with British, Korea could be seen as a useful bastion for G.B to balance French influence in the Pacific and as a step to further creep into the Chinese economy. It could also come in as part of the Great Game as an effort to counter Russian "creep" down from Siberia and nibbling at the edges of Chinese territory. Anglofication is up in the air... the British usually preferred a less culturally intrusive policy in their Second-wave colonial holdings. But its possible.
Spain: The empire had run out of steam by the 19th century. Baring major butterflies, they aren't going to be annexing Korea anytime soon unless they have a few decades of Imperial reform
Dutch: Commerce>political control. Not likely
USA: ... no. Imperial feelings aside I doubt such a war would be popular in the slightest, as unlike the Spainish-American war it would be entirely Asian-focused rather than Carribean-focused (with a Phillipene afterthought) affair. Try selling the idea of American boys dying no closer than halfway across the world to annex a kingdom half the country probably never heard of.