The last thread about this mainly focused on using it a coal station - to enable British control over the Pacific. This could still hold true (and set up for a greater conflict with the Japanese).
I personally think it would be quite important as it gives the British a system of logistics that literally covers the globe. The Hawaiian islands would certainly enable the British to apply force on N.Pacific. Assuming that this doesn't change a lot immediately, the Sandwich Islands may lead to a genuine attempt to bid for Alaska, as a source of coal, and peaceful expansion of British America. Then we have the unknown economic potential of the Pacific Islands. The biggest advantages are more about controlling the trade in the Pacific.
The complications come in with Britain and Russia in the Great Game - Britain has to choose the great rivalry in Asia, or try and ally with Russia to compete with Japan in the Pacific. Allying with Japan may well be a practical option, which could encourage a more liberal Japan that joins the Allies, and cut off WWII in the Pacific, instead freeing the British Pacific forces to help in Europe. Alternatively, neither happens, leaving Britain with an important, but vulnerable Pacific Theater to defend.
The most militant option is that it would make an invasion of Japan from the South and the East more practical. If barely. A British Shogunate or British Japan would be a significant change for the Empire, assuming victory, AND would leave Russia surrounded by Britain and her allies on Every. Single. Front. - Plus, Nanking will be in better straits.