I'm sure that exists as fanfic somewhere (I’ve run Buffy Vs. Godzilla as a one-shot before, with the Big G as the magical summoning of a pop culture icon by an angry sorcerous nerd, following opening Season Five of The Watch House with the more traditional Milli Vs. Frankenstein) but WHEDONesque links to an article in which the existential and humanist subtext of Angel in particular relates to Lovecraftian horror’s cosmic indifference to humanity.
How much of a setting’s underlying philosophy does a reference, an influence or a full-on crossover bring to the table?
I’d say it depends how much the GM and players decide it does, and how much the effect matters to the game as a whole.
The “home” setting probably wins out, if only because that’s the one the players have signed up for. As the old joke goes, the answer to “who would win in a fight...” depends
on whose comic the fight happens in. Or Batman.
It might take it in as an adaptation, be troubled by it as an influence, repudiate it (Angel proves, like the investigators in Call Of Cthulhu, that we do matter, even if “if nothing we do matters... then all that matters is what we do”) or casually shrug it off, depending. Particularly if it breaks one of the fundamental rules of the setting.
Mythos monsters guesting in Doctor Who are just another bunch of pseudo-occult aliens, because the Whoniverse has plenty of those already and has proved repeatedly that humanity absolutely matters. As noted before, the Whoniverse can swallow entire settings whole and solve their fundamental problems in a single episode.
Whereas time travellers are powerful and dreadful things in the Cthulhu Mythos with at most academic interest in humanity, and a mercurial, capricious, charming alien who looks human is probably Nyarlathotep. Although then again, it was always the odd one out among the major Mythos players anyway, because it does take an interest in humans... even if only on the level a cat does with a toy. Hmm. Is the best way to defeat Nyarlathotep to distract it with shiny things?