Possibly, or maybe a closer country such as Japan or another Asian/Oceianic country. Maybe britain would use Australia as a base to reach Hawaii but if it survived to 1914, when World War 1 started (only about 18 years) then it would be taken by Japan as in that time it started to become Island happy. Mabye it became Part of Germany in those 18 years, which would give it even more reason to become part of Japan.
Annexation was not inevitable, but even anti-annexationists in the US would not tolerate any
other country annexing or dominating Hawaii, and that had been true since the 1840's (and of course US interests in the Islands were far greater a half century later):
"The first formal agreement between the United States and the Sandwich Islands, as they were then called, was signed on December 23, 1826. By December 19, 1842, Secretary Webster could assert that his country was 'more interested in the fate of the islands and of their Government than any other nation.' That same year brought a presidential statement opposing all attempts by European powers to annex Hawaii. Since the group lay outside of the Americas, it was not protected by the Monroe Doctrine; but, in terms reminiscent of 1823, John Tyler told Congress on December 20 that if any nation sought 'to take possession of the Islands, colonize them, and subvert the native Government' it could not but 'create dissatisfaction on the part of the United States.' This Tyler Doctrine, as it came to be known, was reiterated at least seven times in the next decade and was supplemented by two related declarations. One was that the United States might feel justified in using force to prevent Hawaii from falling prey to European aggression; the other that the United States would not enter into a joint agreement with England and France to guarantee the *status quo.*"
Richard W. Leopold,
The Growth of American Foreign Policy, p. 131.
https://archive.org/stream/growthofamerican00inleop#page/130/mode/2up
IMO the only realistic alternative to annexation was a quasi-protectorate status vis-à-vis the United States. Other countries could not get a dominant position in the Islands without risking a war with the US--and I doubt that any of them would have been willing to risk it over the Islands.