I always assumed it was fagus to rhyme with magus.
Wood names generally are a minefield where even something that looks as simple as ipe is a trap for the unwary (if you're not familiar it isn't eyepe as in rhymes with Skype, it's two syllables: eepay).
As for the Latin names, we're English speakers so any and all formal pronunciation rules for Latin don't apply, in the same way they don't for Trajan and Caesar for example, which sound very odd indeed if you want to say them 'proper': Trah-zhan and with a K at the front respectively.
Padauk I think most pronounce incorrectly. I've heard native speakers (Burmese) clearly sound a double syllable at the end, so that's how I say it: something like padaook. The dictionary gives it as pəˈdaʊk; -ˈdɔːk for British English which supports this (puh-douk for American English).
The one I had most trouble with personally is sapele, which for years I only knew from reading and never heard spoken. As I thought it was an American wood I pronounced it accordingly. Once you know it's African the correct pronunciation, sapeelee, doesn't sound so weird.
Ilex I would assume is normally pronounced eye-lex by native English speakers (just like ibex). That's British English specifically, there could easily be other norms for our cousins in both NZ or Oz and across the pond, as with the American norm for padauk.
Tilea I'd say was till-eah.
Wenge, more wengay than wengee.
How about Quercus robur? I'd be inclined to say it with -sus at the end, but apparently it's supposed to be -kus (-kəs to be more precise).
