Okay, the only way I can see Japan becoming part of the United Kingdom is if Britain decides to incorporate the Japanese royal family into the British dynasty after conquering the islands.
So, why would they do that?
The best way is to have this as a preventive measure against revolt and rebbelion by the Japanese.
However, this wouldn't really work at all if it was done long enough after the conquest. So what made the British do this?
The easiest way to do that is to have this a lesson from experience. So, what you need is an earlier conquest of a monarchy that led to mass discontent, and preferably, the ejection of the ruling party, eg. the Brits.
As a colonial method, the previous lesson must have occured in the colonial world, as opposed to Europe. Africa and the Americas don't have any countries that fit this example, so it's got to be Asia.
So, let's say a wave of Asiaphilia overcomes Britain sometime in the mid-eighteenth century, and Britain attacks and quickly overcomes an Indian or SE Asian state (make it bigger than any previous gains), but is ejected by rebellion.
So now we have a reason for Britain to absorb the Japanese monarchy, but not a reason for them to colonize Japan.
How's this: The wave of Asiaphilia was not restricted to Britain. A rivalling colonial power (Spain works best in this case) conquers a near-to-Japan country, say, Korea. In order to counter Spanish influence in East Asia, Britain 'protects' and occupies Japan. The Spanish are eventually kicked out of Korea, but because of the union, Japan stays British.
Let's say this all happens between the Seven Year's War and ARW. We can assume that there's still enough discontent and rebellion in Japan to take away British troops from the ARW, so we might see Canada becoming part of America, or the Americans winning earlier.