If there had been a US-UK war in the 1840's over Oregon--not likely, but possible if Cass had been elected president in 1844--I could see the British seizing Hawaii and making it a protectorate. Otherwise, it was just not worth the bad effect it would have on British-American relations.
In OTL, the closest the British came was Lord Paulet's seizure of power in 1843, but this was soon repudiated by the British government. Lord Paulet did have his supporters on the Honolulu waterfront, who rejoiced at seeing the psalm-singers lose their influence--among other things, Paulet relaxed the liquor laws of Honolulu and repealed the laws against fornication (except in public streets). [1] Herman Melville, clerking in Honolulu at the time, strongly defended Paulet's conduct. However, as J. C. Furnas sums it up in *Anatomy of Paradise: Hawaii and the Islands of the South Seas* (pp. 154-5)
https://archive.org/stream/anatomyofparadis012497mbp#page/n169/mode/2up "Unwilling for ports so important to American whalers to be in British hands, Washington protested sharply. John Bull was already nervous about Uncle Sam *in re* Texas and Oregon. But, before instructions could reach the Pacific from England, Rear-Admiral Thomas, Paulet's senior, arrived at Honolulu in HMS 'Dublin' and formally repudiated his lordship's doings. The king was reinstated in a ceremony still commemorated by the name of Thomas Square in Honolulu." It was during that ceremony that King Kamehameha III famously said "Ua mau ke ea o ka 'aina i ka pono" ("the life of the land is perpetuated in righteousness"--which to this day is the official motto of the state of Hawaii. Admiral Thomas remained in Honolulu until March 1844, "suffering as best he could through more temperance dinners..."
http://books.google.com/books?id=w19C8zZC21EC&pg=PA117
[1] The Hawaiian Kingdom at the time of Paulet's seizure had been heavily under the influence of US missionaries. Note this provision in the Kingdom's 1840 Constitution: "That no law shall be enacted which is at variance with the word of the Lord Jehovah, or at variance with the general spirit of His word. All laws of the Islands shall be in consistency with the general spirit of God's law."
http://www.hawaii-nation.org/constitution-1840.html Someone in soc.history.what-if once proposed as a challenge getting this language (apart from the words "of the Islands") into the US Constitution...