Why does American English still write "glamour" with a "u"?
Solution 1:
Because it was not a French word, but a Scottish one. And we did lose a u — just not the u you were expecting.
Per the OED, it was a corruption of grammar, which during the 18th century was variously spelled glamer, glamor, glammar, and then in Scotland, as glaumour. That was one u too many, though, and it went then to glamour where it has remained ever since.
Then during the 19th century, Sir Walter Scott started using it in his literature, and it became popular throughout Anglophonia. The OED says:
Etymology: Originally Sc., introduced into the literary language by Scott. A corrupt form of grammar; for the sense cf. gramarye (and Fr. grimoire), and for the form glomery.
Its original sense was a spell, an enchantment, a dweomer — in effect, a charm. Its glamorous sense of fashion charm came later. There are lots of derived terms like glamour boy, glamour girl, glamour gift, glamour-learned, and of course, glamour puss.
Please see also the related question How and when did American spelling supersede British spelling in the US?, which mentions not only the historical spelling change for this word but for many others. The changes happened at different times to different words — and to this one, not at all.
Solution 2:
The reason the spelling wasn't changed appears to be that Noah Webster didn't know about it.
The word glamour does not appear in the original 1828 Webster's Dictionary, so he couldn't change its spelling in that dictionary the way that he did for armour, honour, humour, neighbour, etc. In fact, it does not even appear in the 1890 Webster's High School Dictionary, which is abridged from the 1890 Webster's International Dictionary. This is fairly surprising because Ngrams shows that the word was fairly widely used in the U.S. by then.
Solution 3:
Whether Noah Webster knew the word glamour or not, he didn't include it in any of the Webster's dictionaries published during his lifetime. This circumstance is somewhat surprising given that, according to Merriam-Webster's Eleventh Collegiate Dictionary (2003), glamour first occurred in published English writing in 1715.
Webster published three major editions of his dictionaries: A Compendious Dictionary of the English Language (1806), An American Dictionary of the English Language (1828), and An American Dictionary of the English Language, second edition (1840). In none of these does glamour appear. The Compendious Dictionary consists of entries rarely more than a line long and owes a huge debt in wording to Johnson's dictionaries. The 1828 American Dictionary provides detailed definitions and many illustrative quotations, as does the 1840 American Dictionary, which is an update, not a rewrite, of it predecessor.
Webster instituted his simplified spelling of certain words (such as words ending in -our) in the Compendious Dictionary, and continued to enforce those spellings in his subsequent dictionaries. I have no doubt that he would have simplified glamour to glamor if he had thought to include the term at all. But he omitted it, even from the 1840 dictionary, which, on page 377, skips from GLAIR to GLANCE. (The 1846 printing of this dictionary, "abridged from the quarto edition of the author"—which is the edition available for viewing online—includes "An Appendix containing all the additional words in the last edition of the larger work" of the 1840 quarto edition. In this appendix, on page 965, the list of words skips from GLAIRED to GLANCED. Glamour and glamor are nowhere to be found.)
Interestingly, Joseph Worcester, Noah Webster's former assistant and (later) chief antagonist in the "dictionary wars," omits glamour from his dictionaries as late as A Comprehensive Pronouncing and Explanatory Dictionary of the English Language (1860/1874), indicating that the word simply wasn't very widely known and used in the United States as late as the 1860s.
Soon after Webster died, the G. & C. Merriam Company acquired publishing and revision rights to Webster's work. As the back-cover copy to a 1970 facsimile reprint of the 1806 Compendious Dictionary puts it,
Following Noah Webster's death in 1841 [actually, as Tonepoet notes in a comment below, Webster died on May 28, 1843], George and Charles Merriam obtained exclusive publishing rights to Webster's American Dictionary copyrighted in 1840, which was Webster's last and most comprehensive work, together with exclusive rights to publish revisions and abridgments. They published the first Merriam-Webster dictionary in 1847.
That 1847 edition of An American Dictionary of the English Language includes the following entry, between the entries for GLAIVE and GLANCE:
GLAMOUR, n. Witchery, or a charm on the eyes, making them see things differently from what they really are. {Scottish.} [Example:] It had much of glamour might/To make a lady seem a knight. Scott's Lay of the Last Minstrel [1805].
Webster's New University Pronouncing Dictionary, published by J. B. Lippincott & Co. in 1856, undoubtedly to the chagrin of G. & C. Merriam, has a similarly phenomenological definition of glamour:
GLAMOUR n. A magical deception of the eyes, making things appear different from what they are.
Glamour appears in all unabridged editions of Webster published by G. & C. Merriam thenceforth. In the 1864 edition of An American Dictionary of the English Language, the entry is unchanged from the one that appears in the 1847 edition. But in Webster's International Dictionary (1890) the entry grows to include four distinct definitions and some additional terms:
Glamour n. {Scot. glamour, glamer; cf. Icel. glámeggdr one who is troubled with the glaucoma(?); or Icel. glām-syni, weakness of sight, glamour; glāmr name of the moon, also of a ghost + syni sight, akin to E. see. Perh., however, a corruption of E. gramarye.} 1. A charm affecting the eyes, making objects appear different from what they really are. 2. Witchcraft ; magic; a spell. Tennyson. 3. A kind of haze in the air, causing things to appear different from what they really are. [Example:] The air filled with a strange, pale glamour that seemed to lie over the broad valley. W. Black. 4. Any artificial interest in, or association with, an object, through which it appears delusively magnified or glorified. Glamour gift, Glamour might, the gift or power of producing a glamour. The former is used figuratively, of the gift of fascination peculiar to women. [Quotation from Scott's Lay of the Last Minstrel omitted.]
For its part, the adjective form glamorous (without a u in the second syllable) makes its debut in Webster's New International Dictionary (1909).
But this begins to take us too far afield. The short answer to the poster's question is that glamour became the standard U.S. English spelling of the term because it escaped Noah Webster's clutches during his lifetime, and Webster's successors didn't apply his systematic, prescriptivist alteration of -our words to -or to the previously unacknowledged word. Evidently that zeal died with him.