⟨e⟩ almost certainly always represents [ɛ], not [e]. Appendix E of The
Lord of the Rings describes it as the sound in English “were”, and I’m
not aware of any English dialect that pronounces “were” with an [e].

In Quenya, long ⟨é⟩ (and, presumably, ⟨ê⟩) is, according to Appendix E,
“tenser and ‘closer’”, which I assume means [e]. Several online sources
agree with this as well.

In Sindarin, Appendix E is quite clear that ⟨é⟩ has “the same quality”
as ⟨e⟩, only differring from it in length: I assume this must mean that
⟨é⟩ is [ɛː] in Sindarin. The online information on this is confusing and
sometimes contradictory even within the same page; several sources claim
that Sindarin has an [eː], but I have not seen this claim substantiated
with a source from Tolkien, and I suspect it’s simply a confusion with
Quenya. It scarcely matters, anyway: Sindarin words with ⟨é⟩ or ⟨ê⟩ seem
to be pretty rare. (I’m aware of a single word with an ⟨é⟩ – the name
Eluréd, son of Dior – and the Omikhleia Sindarin dictionary [1] features
some words with ⟨ê⟩, giving their pronunciation with [ɛː].)

The [[EI]] phoneme for Sindarin ⟨ei⟩ is copied from the base2 phonemes.

[1]: https://www.jrrvf.com/hisweloke/sindar/index.html