The Fey origin works well for the British elf, who are explicitly faerie.
D&D has an awkward disconnect between Fey and Elemental. Both would be nature beings, and both would be primal power source.
The Elf could be Elemental. The British elf is a land spirit associating with the plants. The Norse elf is a sky spirit associating with the sunlight, which feels even more Elemental.
One solution might be that all of it is aspects of the Ethereal Plane. The ether includes the four elements, and also includes the fey and the shadow. Potentially some earth elementals might be fey land spirits, including Gnome.
Altho a land spirit, the British sith elf might specifically be a Fey Plant Elemental, relating to fertile soil. And altho a sky spirit, the Norse alfar elf might specifically be a Fey Fire Elemental, relating to sunrays.
If the Feywild itself includes Elemental beings, the aspect would be the four elements coexisting with each other, in a dynamic equilibrium that brings forth and sustains life.
(This elemental eqilibrium compares to the Dark Sun concept of preserver magic where the four elements are in a lifegiving harmony. Perhaps Athas became unable to access the Feywild because it became too out of balance because of the destruction of water.)
Similarly, the Norse Dvergar dwarf would be a Fey Earth Elemental.
The Norse alfar and dvergar are Fey in the sense of personifying human fates, successful and unsuccessful, respectively. But they retain their elemental aspects, respectively.
I feel comfortable relating the alfar with D&D both Fire and Air, and the dvergar with both Earth and Water. There are a number of Norse texts that correlate this.
If each elf correlates with two elements, then the British elf is definitely both earth and plant. Whence the D&D Wood Elf is earth and plant, and the High Elf might be plant and air. The Drow Elf is obviously earth, possibly water too, thus correlating but nonidentical with the Norse Dvergar.