I see that Terry Teachout is writing an essay on Edward Elgar, and "trying to make sense out of the peculiar fact that his music has never been popular outside England." Put me down as another Elgar fan who is not from England (I'm from Canada, but that hasn't counted as English since at least the '60s). I've loved Elgar's music ever since I heard Adrian Boult's recording (with the London Symphony Orchestra) of Elgar's most popular work, the "Enigma" variations. And I too find it unfortunate that Elgar's music isn't better-known outside of England. The Variations are well-known enough, the Cello Concerto is a best-seller thanks to the late Jacqueline Du Pre, and everyone who's been to a graduation has heard the first Pomp and Circumstance March, aka "Land of Hope and Glory" (though the second march, in A minor, is probably better -- more wistful than pompous or jingoistic). But Elgar wrote many other masterpieces: the "symphonic study" Falstaff, the only musical work that successfully portrays the multilayered Falstaff of Henry IV instead of the dumbed-down Falstaff of Merry Wives of Windsor. The Introduction and Allegro for strings, an unusual combination of accessible and "difficult" elements: beautiful tunes alternate with complex fugal development (or what Elgar himself referred to as "japes and counterpoint"). The oratorio The Dream of Gerontius, the Second Symphony, the Violin Concerto, the String Quartet... so many great works, and all in a musical style that's unique to Elgar.
I wish I were enough of a musician to describe his style; one site describes it as "a synthesis of European influences, particularly Brahmsian structure and counterpoint, and Wagnerian harmony, with a uniquely English nobility and grace." That's true enough, and one of the things that makes Elgar's music so interesting is that he combined the influences of Brahms and Wagner at a time when those composers were thought to represent two opposing camps (some musicians were "Brahmsian" -- favouring traditional symphonic structure, absolute music, counterpoint, etc; others were "Wagnerian," favouring programmatic music, and privileging advanced harmony over rhythm or traditional structure). But one of the things that makes Elgar so fascinating is that you sometimes don't know, from moment to moment, what mood he'll be in or what side of his style will be predominant. I've already mentioned how the Introduction and Allegro swings back and forth between attractive tunefulness and learned fugues; but perhaps the great Elgar moment comes in the third movement of the Second Symphony, where what starts as a charming, playful scherzo gives way to a "relentless beating" that drowns out the rest of the orchestra and, done properly, can actually scare the audience; the effect was described by Elgar as "similar to that dreadful beating that goes on in the brain" of someone with a high fever. That kind of emotional range is what makes Elgar such a wonderful composer; and the Enigma Variations are a good place to start with his music, because the variations (which are musical portraits of Elgar's friends) show such a wonderful range of moods and styles, all within the variation form. Good recordings include the Boult recording mentioned above, or the recent recording by Mark Elder (who is doing a very fine series of Elgar recordings with the Halle Orchestra).
Outside of England, I think, Elgar is often seen as the embodiment of parochial Englishness, the composer of hymns to Empire like "Land of Hope and Glory" and of the smugness of the Edwardian era. This attitude was actually prevalent in England for some years before and after Elgar's death (in an obituary, the Musical Times expressed the hope that "a few years hence the association of certain works of Elgar’s with the Imperialism of thirty years ago will be no more than a vague memory"), but the English grew out of it. The rest of the world, not so much; and Elgar is still often seen as the musical symbol of everything people don't like about the English.
The irony is that Elgar was the most non-parochial English composer up to that time; he was the man who proved that English music could be "international," and he was accused, if anything, of not being English enough. England had never produced a serious composer of true international stature since the death of Purcell (Sullivan certainly had international appeal, but not for his "serious" works). The prominent English composers of the Victorian age tended to be guys like Stanford and Parry -- people who tried to write in a specifically "English" style, ignoring international influences as much as humanly possible. Elgar would have none of that; his music showed off the influence of Brahms, of Wagner, of Richard Strauss. Unlike many of the English composers who came after him, he didn't have much interest in English folk song or the madrigal or anything else native to England. He wrote music that was of the world, not just of England -- and by doing so, he won international popularity for many of his works, particularly the Enigma Variations. Composers like Benjamin Britten, who also used a melange of styles and didn't worry about including nationalistic elements in their music, owed a debt to Elgar (as a conductor, Britten made probably the greatest recording of the "Introduction and Allegro," included on the World of Elgar CD linked to above). Shaw, who as a music critic repeatedly blasted the English musical establishment for its parochialism, recognized that Elgar represented a new hope for English music, and championed his work.
But maybe that very "internationalism" is, ironically, one of the factors that contributed to Elgar's declining popularity with an international audience. That mixture of international styles, that sort of Germanic Englishness, has sometimes struck non-English critics as representing the failed attempt of an English composer to imitate his "betters" on the Continent. (One American critic sneered that Elgar was an inferior Anton Bruckner.) More importantly, despite the emotional range I mention above, Elgar rarely "cuts loose" -- there's always a certain restraint, no matter what mood he's in; there are few moments of wildness or sheer weirdness like with his contemporary, Mahler. That restraint may be the most typically English thing about Elgar's music, and it may be part of what keeps him from being big with North American audiences, who tend to expect late-romantic music to be a little crazier.
Anyway, that's my two cents -- well, given the length, at least three cents -- on Elgar. Looking forward to the essay.