...in terms of the establishment of a British-Japanese monarchy, I highly doubt any union COULD occur apart from TheNordicBrit's Mongol possibility could occur. The more likely option is either Dutch or Spanish domination. The Japan of the Nobunaga-Hideyoshi-Tokugawa period was a time when the western world was first in contact with Japan, particularly the Dutch and Spanish (who had a large presence in Java and the Philippines, respectively). Totoyomi Hideyoshi had initially wanted to establish Christianity as a State Religion--until he realized that the Spanish were creating base points in all catholic-based areas. One of the main reasons that Christianity initially was banned was that some of the early Catholics in Japan had in fact attempted to request three battleships from Spain. Sir Francis Xavier, the first missionary to Japan, had considered Japan a launching point for an invasion of China. If the letter had not been intercepted by Tokugawa (or if the remnants of the Hideyoshi family had defeated Tokugawa), it may not have been unreasonable that Spain may have come to rule Japan. After the death of Hideyoshi, the Hideyoshi family represented a symbol of anti-tokugawa sentiment (which included Ronin, Western Powers--and, particularly, the Christian-Japanese Population), and if the Hideyoshi family had triumphed, its scion could hardly have maintained control over a divided Japan, establishing a power vacuum that, through marriage or political activity, could eventually lead to a union between the Spanish Crown and the Japanese.
...just sounding off ideas here, I'm sure there are quite a few timelines with this info...