Why, in US English, does the word glamour retain its u while humour, neighbour, and others have shed it?
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:
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.
The reason the spelling wasn't changed is 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 1892 Webster's International Dictionary, which is fairly surprising because Ngrams shows that the word was fairly widely used in the U.S. by then.