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.
I tend to drop the Ethereal plane, but I think that Eberron hints at a good way to go about the Feywild and potentially the Shadow Fell. Well, they at least have the direction.
The Feywild is not only life plus, it is a realm of dreams made reality. The Shadowfell is nightmares made reality. I think the Sorrowsorn highlight this the best. The Hungry is a creature that ( in some versions of the lore) comes about when a mortal in the Shadowfell is starving and obsessed with hunger. Meanwhile, in Faerie, you would have something similiar in a creature that is constantly feasting, and draws others into participating in an endless feast and party. It is less about the elements and more about the emotions.
The problem is creatures like Dryads. The feywild is also where we find the nature spirits. But, thinking about it... I'm not sure how much sense that actually makes.
Pivoting, the elemental planes are pure expressions of the elements. Water, Earth, Fire, Air. They aren't a source for plants and animals. You can have plants in a purely water sense, with plankton, but pure earth isn't conducive to plant growth. Plants and Animals are expressions of the Prime. I'd almost consider it more likely that plants and animals migrated from the prime back to other planes, while the raw stuff that made the Prime came from those other planes.
This actually ties into why I prefer the Elemental Chaos to the four planes. The only usage I can think of for the four planes is to make them the raw stuff that was used to make the Prime. The only planes that have any real major civilizations I've seen are the Water and Air planes. Fire has the City of Brass, but that is a single city. Water has... so much. But, I could easily see transferring it to just be in a distant ocean. The Oceans of the Prime are vast and generally underutilized, and the Elemental Plane of Water is just Ocean plus so I could see moving some of that into the Prime with little disruption.
So, how does this tie to Dryads? I think maybe you steal an Eberron concept. In Eberron the Rakshasa and a few other inhabitants of Khyber are "native fiends" they are fiends native to the Prime, whereas something like the Balor or Pit Fiend is an Outsider, a fiend from the plane of Fierna. Dryads are fey, but they are fey native to the Prime. They are less a dream and more an extension of a dreaming plant. A Naiad would be the same way. Not a fey of the Feywild, but a fey of the river's dreams.
Does the feywild have Dryads and Naiads? I think so, but I think that is because the Feywild is a reflection of the prime, and therefore it has things from the prime in it. I'd also then say we would have Dryads and Naiads in the Shadowfell. The hungry spirit of a hangman's tree would be a type of nightmare Dryad.
I haven't developed this idea at all, it just came to me while reading your post, but I'd be curious if it has legs.