This is “a voiceless w, as in English white (in northern pronunciation)”
according to Appendix E of The Lord of the Rings, and so we copy the
[[w#]] phoneme from the English phonemes. I can’t actually hear much of
a difference from the previous [[hw]] (I know what the difference
between [[w]] and [[w#]] should be, but [[hw]] already sounds like
[[w#]] to me), but at least this improves the --ipa output, changing it
from [hw] to [ʍ].

I’m not aware of any statement by Tolkien to this effect, but it just
seems natural to me, and the [ŋ] is at least found in the Omikhleia
Sindarin dictionary [1] (e.g. entry tinc, [tˈiŋk]).

Based on a similar phoneme rule in ph_english.

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

For ⟨ae⟩, we used to use the Latin [[aI]] phoneme, which sounds like
/ae/. Call that phoneme [[aE]] for us and introduce a separate [[aI]]
one, which sounds more like /ai/ (since Sindarin has both, and they’re
supposed to sound different, though Appendix E of The Lord of the Rings
notes that there is nothing closely corresponding to ⟨ae⟩ in English and
that it may be pronounced like ⟨ai⟩). Furthermore, for ⟨oe⟩, just remove
the TODO – the Latin phoneme is called [[OI]] but sounds more like /oe/
than /oi/, so it’s actually just fine for our purposes. Finally, the
⟨ui⟩ diphthong is copied from Finnish, just like in Quenya.