As an American, I noticed Fluorine and Iodine, though clearly sharing the same ending (and this was corroborated by etymonline.com; both contain the same chemical suffix) sounded different. Fluorine has an ending that sounds like "een", and Iodine has an ending that sounds like "ine" should sound if you say it aloud. Bromine, Chlorine, and Astatine also have an "een" ending, so clearly Iodine is an outlier. I then went on en.wiktionary.org to check it out, and it turns out the British pronounce it correctly, whereas the Americans do not. I searched all my usual sources and some more, but nothing could explain this difference. Any help?
I don't think there is any good explanation.
I checked this in Fowler (originally 1926; re-published 2009). He writes that iodine, bromine, chlorine and fluorine were at the time all given with /ɪn/ as the main pronunciation in the OED, with /aɪ/ as an allowed alternative. He also thinks that "iodine" generally had /aɪ/ in popular usage at his time, unlike the other three.
Apparently the prevalence of different pronunciations has shifted since his time. As you note, the pronunciation with /iː/ seems to be common in modern British English (the OED now lists it first). In general, words ending in the suffix -ine have very variable pronunciation, unrelated to any etymological distinctions (e.g. divine, bovine vs. marine vs. genuine). It's somewhat comparable to the variability in the pronunciation of -ile. Although some, such as John Walker, have attempted to prescribe /ɪl~l̩/ vs. /aɪl/ on the basis of Latin vowel length, it seems quite arbitrary to use this as a criterion of correct pronunciation today, especially since /aɪ/ is actually stigmatized in the pronunciation of the word genuine which is from Latin genuīnus (and as far as I know, /aɪ/ is unheard of in doctrine, from Latin doctrīna).
Another oddity is that "machine", which in Latin had a short vowel in the second syllable and was stressed on the first vowel (māchina), is in modern English pronounced with stressed /iːn/ in the last syllable due to coming through French. (This word doesn't have the same suffix as the other -ine words, but it ends in the same letters.)