I don't have a definitive answer to this question and it is not something that could ever be really scientifically validated. Nevertheless it's the sort of question that no-one has an answer to and draws my curiosity. I was prompted to consider it after reading, somewhere, a theory that the peculiar English sound of Elgar was due to the way the drooping melody lines match certain patterns of English speech. I thought this was nonsense because no one would be able to pick up such a connection, even subliminally. I mentioned this theory to a friend when we were both sat in a room with Elgar playing and he thought likewise; "nah, it's because it sounds....pompous". Just how is it that Elgar's music has this particular sound that resonates so much with people in England, whilst not sounding this way to any other country?
When we hear music containing national instruments, like bagpipes, we think of the country they are from but this is purely by association. Elgar, on the other hand, managed to create a musical style that innately captured a national spirit, and only the British perceive it as such. Many Russian composers sound Russian to everyone but this is possibly because there is a strong Russian musical tradition, and Shostakovich sounds 'Communist' and hence Russian because his music frequently sounds like a military parade.
Elgar, however, achieves a nationalistic sound without any external references and it is purely down to the musical style.
Whilst studying music one my lecturers, who specialised in composing tuneless "modern" music, had a particular dislike for Elgar because it was, he considered, so backward and conservative in style. I find such a comment utterly ridiculous. Elgar was, in my opinion, one of the most original composers of all time, ranking up there with any of the stylistic pioneers of modern music. I think the Enigma Variations is as original as The Rite of Spring. After 150 years of there being no major English composers, and in a musical scene dominated by Germany, he created, from scratch, a new sound and style that perfectly captured the mood and sentimentality of an entire nation. That's pretty impressive, far more so IMO than creating a load of disjointed harmonies and time signatures of The Rite of Spring. The brilliant musicologist, Wilfrid Mellers, in Man and His Music, summed up Elgar's originality: "it took a genius to manifest in music the spirit that justified apparent banality, even brutality, of thought and feeling" (ie the harsh excesses of the British Empire at its height).
In fact, Wilfrid Mellers comes about as close as possible to defining why Elgar sounds English without actually nailing it: "The originality is perhaps more melodic than harmonic. Elgar's lyricism has the seething energy of Strauss's early symphonic poems, but the curves are more rounded. The surging-upward figures on the strings...gives to his most exuberant phrases a relaxed ampliture.", "we find a sensitive melodic line, humanly intimate rather than rhetorical" and "in the free rubato of the lyricism an intimate human voice speaks directly to you and to me, while an unexpected chord or modulation reveals the private heart beneath the public manner". There is also a discussion of Elgar's harmonic style which seemingly little different from the German musical mainstream of the time has subtle originalities.
What, then, is so English about Elgar?
Taking a fresh look at Elgar, one thing that becomes apparent are his somewhat throwaway melodic lines. It occurred to me just how quirky some of his tunes are. Take the first part of the Pomp and Circumstance March No1, for example, and all of March No2, and the 3rd movement of the cello concerto; the tunes are humourous, light-hearted and almost cheeky in style. His slower themes, as Mellers points out, have rounded endings and are lighter and more lyrical than the Germanic musical mainstream.
And yet the orchestration is NOT humourous, light-hearted and cheeky, but loud, bold and almost pompous.
With these factors in mind, I think it is not far wrong to state that the key element of Elgar's Englishness is in the way he uses cheeky tunes in the fast bits and overly-sentimental lyrical tunes in the slow bits, but wraps it up in a serious and brash orchestration. It is in the contradiction between the light-hearted melody and the definitely-not-light-hearted orchestration. One of the abiding characteristics of the English is in not taking ourselves too seriously, "lightening-up", and disavowing strong displays of emotion. Elgar appeals to us because Elgar's melodic lines that resonate with our character are made proud and bold by his orchestration. In other words, the light-hearted tune that resonates with our light-hearted attitude is put on a platform of grandeur and as a result the English laid-back spirit becomes ennobled. This suits us because we do tend to think of our sense of humour as rather superior. And though Elgar can be very emotional it is presented more subtlely with a more flowing and melodic orchestration compared with his contemporaries. In the loudest parts the edge is almost taken off the emotion by the orchestration and this goes hand in hand with the English resistance to getting too emotionally explicit, in public at least.
Elgar's 'English' sound is a consequence of his light-hearted melodic style that is proudly proclaimed with bold orchestration, rending that very light-heartedness something that has real depth and meaning. English light-heartedness is elevated to a platform of grandeur. Perhaps the reason his music seems 'pompous' is because the orchestration is almost too intense for his tunes. The orchestration seems overly proud and assertive - and hence pompous. His spirit of light-heartedness has much in common with Gilbert & Sullivan and later composers of light music such as Eric Coates, music which also sounds very English. Elgar's style can be described as "serious light music" very much in the English tradition of light music, but with Germanic symphonic grandeur.
Why Vaughan-Williams also sounds English is a question for another day.