Funny you should mention Elgar. He is, in a sense, my local composer- we are both Worcestershire lads…
In his day, he was the British composer closest to the European (German) Romantic tradition. His Cello Concerto is a staple of the repertoire, alongside the Dvorak and the Schumann…
You absolutely must get hold of the Du Pré / Barbirolli (if you don’t have it already). This sets the standard musically, and is a first-rate audiophile recording. Also please check out the violin concerto, the second symphony (Boult), or, for a real system test, The Dream of Gerontius (Boult or Rattle).
The Du Pré / Barbirolli is my go to recording as well, but I do not have it on vinyl. It's funny you should mention that as I was talking of buying a copy just yesterday thanks for the other recommendations, I will surely check them out.