The primary reason that Hawaii was annexed as a territory of America is that they believed (and rightfully so) that if they didn't annex it, some other nation would, but which nation? There are a variety of possibilities, from the ever expanding English Crown, to the Japanese empire. If for some reason, America was unable to annex Hawaii, what nation (if any) would most likely annex it?
When I was on vacation I bought a little book on the history of the islands. The economics and population make this answer obvious. The population was evenly split at 1/3 from the US (caucasian), 1/3 Japanese, and 1/3 Polynesian. The economy was based on US and Japanese interests. Why ever would England get involved?
I don't believe your opening sentence, either. Businessmen influenced the politics and US had close economic ties for some time; only reason it wasn’t annexed earlier was prejudice over not having a white majority population.
The US had a military base there since 1887. It became a territory in 1898.
If another country were to become tied, it would have to be before this time. Read the Wikipedia article and consider what would happen if the US was not expansionist or didn’t have the resources at the moment or other political issues got in the way: if the business interests wanted to stabilize the government (read: overthrow the monarchy) with help from gunboats, if the US was not supplying one then Japan would be the next choice.
- New Zealand (then part of the British Empire) => they already left Hawaii, so obviously didn't annex it
If I look a the other islands in the Pacific, all of them either seem to belong to France, to New Zealand, or to be independent (i.e. Micronesia). Japan does not seem to have any far out islands in the Pacific.
Since no one else has mentioned it (and to add some variety), what about Germany? Or more realistically, a German version of the East India Company seeking to "stabilize" the area in exactly the same manner as the U.S. did. A German "entrepreneur" just does it first. I'm not saying this is the most probable, just putting it out there as a possibility.
England was not expanding her interests in the Pacific at that time, Japan is the obvious choice — but they were not really active in grabbing overseas territory until about 1894 when they started grabbing bits of China — if the US had held off until the early 1900s they would have faced a challenge from Japan I think.
Interestingly Japan was a key British ally in WW1 and Britain may have had a tough time choosing between supporting Japan and Supporting the USA if there had been an issue over Hawaii in 1914.
At least in part, I think your reason why the US can't/won't annex Hawaii will play a role in this. Prior to the Spanish-American war (1898), Spain had possession of Guam and the Philippines. If your alternate history precludes the US as a military power, they may have kept them, or even expanded to rule other islands as well.