The Battle of Hawaii
Well, the American Marines had one hell of a job to do. The Marines had disarmed most of the native forces and armed many of the pro-US settlers. Amusingly, most of them were actually former Confederates, including a few veterans of the US Civil War. Many of the natives were outraged, but lacking arms after the surrender of the Hawaiian Royal Guard and the small Anglo-Qing garrison, they had no method of resisting.
The US Marines were also escorted by the only American pre-dreadnought not on the East Coast, the USS Iowa, the most advanced battleship in the American arsenal (the other four battleships were members of the Indiana-class, which were primarily designed for coastal defense, the job that they were handling right now). That being said, three battleships from the undefeated Royal Navy were sailing towards Hawaii. Admiral George Dewey, stationed on the USS Iowa, led the Royal Navy on a wild chase around the islands of Hawaii. The Hunt for the USS Iowa delayed the British landing by several weeks, though eventually a confrontation was forced, at which point the USS Iowa was sunk off the coast of the Big Island (albeit taking down two British cruisers with it). Ironically, the sailors of all three ships had to swim to the same island, at which point they declared a cease-fire against each other, because there was honestly just no point in fighting anymore there.
The real confrontation would be in Oahu, still under American occupation. Although two Qing cruisers had arrived as support, the actual landing force would be primarily British, largely but not entirely recruited from the British Raj in India. Although troops from the Australian crown colonies had been raised, those had largely been directed to Aljaska/Canada as they were raised before the confrontation with America. The British landings focused primarily on Ewa Beach, just west of Pearl Harbor, Lanikai Beach on the windward coast of Oahu, and Waikiki Beach right in Honolulu.
The Waikiki landings were an absolute disaster. The Americans had set up large amounts of artillery in the Koolau Mountains and had a tendency of shifting their artillery after a few shots, making it impossible for British battleships to properly estimate their location. A successful landing in Waikiki would have endangered the Iolani Palace, which the US Marines had turned into their HQ. Taking the Palace would have been a huge win for the British, as the conquest of Hawaii was largely accomplished by US troops storming the Palace. However, in the wide open beaches of Waikiki, American-dug fortifications and machine gun emplacements (using captured Maxim guns!) just massacred landing troops. British battleships were unable to support the landing out of fear that the shelling of Honolulu (the most heavily populated part of Hawaii) would alienate the natives from British influence.
In contrast, the landings in Kailua were largely successful, as American forces were concentrated on the leeward coast of Hawaii. However, advancing outside of Kailua was difficult, as it was separated from the rest of Hawaii by the Koolau mountains, where the Americans had put most of their artillery. Attempts to move to the rest of Hawaii, alongside fairly narrow roads alongsides the coasts, ended in total disaster as British troops were simply sitting ducks for American artillery and harassing attacks.
As such, the main British advance would truly have to be on the Ewa Beach, where they had both the benefit of easy support, and enough distance away from the Koolau Mountains to minimize American artillery support. After a fairly successful landing, British forces settled into a grueling siege of Pearl Harbor, confronted with the fact that the Americans had built all kinds of trench-works blocking the road leading to the Harbor. An attempt by British troops to circumvent the Americans by trying to go around them failed as the British forces were totally unprepared for warfare in the rice paddies in that region, while the Americans were bizarrely overenthusiastic about fighting in swampwater (many of them were Southerners, disavowed by their home country, very used to the swamp). As a result, the strategy quickly became for British battleships to slowly create rolling barrages into the American earthworks, while British troops advanced slowly. Although the British were able to fan out and occupy much of the North and West of the Island, their advance into Pearl Harbor was steady...but excruciatingly slow due to extensive American defensive preparations. In addition, even in the "liberated" parts of the Kingdom of Hawaii, the difficulty of dealing with hostile, armed, and very skilled settler-guerillas was obvious, as they refused to meet the British troops in open combat, simply fighting them in assymetric warfare. The natives generally supported the British and provided labor and logistical support, but generally lacked experience in industrial warfare, so the British still had to do the heavy lifting.
As most of the pre-existing support staff was actually from Qing China (such as "coolies"), the British forces in Hawaii were interestingly remarkably multicultural, including native Hawaiians, Chinese of all stripes (both Han and Manchu), Indian soldiers, and British officers, which stood in contrast to the American forces, who were a mix of both USA and CSA nationals (the latter disowned by their home country, which pledged neutrality in the conflict).
