I was surprised to find that what I thought was modern usage is first reliably attested in 1941, though the unattested sense is a bit earlier. Just thought I’d share.

What commonly used words meaning homosexual are politically correct in the US and elsewhere?

I’m not sure what you mean by the ‘unattested sense’ being earlier, unless you mean cases where one can’t quite be sure. The OED gives a few examples of antedates where it hedges its bets. Of them, the Coward example from 1939, where it’s combined with ‘queer’ (attested in the homosexual sense from 1914, and which I’ll hazard a guess was the most common term of the day, though of course it also had its more general sense), seems to be the best candidate.

The Coward citation from 1929 is also pretty clearly in the sense of homosexual. The “green carnation” is s clear reference to Oscar Wilde.

1929 N. Coward (We all wore) Green Carnation in B. Day N. Coward: Compl. Lyrics (1998) 114/3 Art is our inspiration, And as we are the reason for the ‘Nineties’ being gay, We all wear a green carnation.

The Bringing Up Baby citation from 1938 is not a clear use in this sense. Ron Butters makes a persuasive case that this is not a use of the slang sense of homosexual. (I have to update the Big List entry on this one.)

The candidate citations that I’ve seen that predate 1929 are almost certainly not used in this particular slang sense, although they often seem so on the surface. The problem is that in the opening decades of the twentieth century gay had a number of sexual or hedonistic senses, and determining exactly what was meant in a particular case is often impossible. That’s probably why it made such a great code word for the homosexual community; it provided plausible deniability if overheard or directed at the wrong person.

(My use of homosexual here sounds really bad. Normally I’d write things like “the gay community,” but I’m trying to avoid using the word gay to avoid confusion.)

What commonly used words meaning homosexual are politically correct in the US and elsewhere?
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I’ll speak for part of elsewhere (Australia). Some commonly used words meaning homosexual that are politically correct in Australia are homosexual, gay and queer.

Is “queer” politically correct in Australia even when used by a heterosexual?  In the US, it’s been adopted as a defiant self-indentifier (e.g., “We’re here, we’re queer. Get used to it.") but would be very non-PC and inflammatory if used by a heterosexual in reference to a homosexual.  A close parallel is the use of the “N-word” by blacks.

Queer is commonly used in literary theory and criticism. You talk about “queer literatue” or “queer theory,” but I suppose that’s an application od the reclaimed word, even if by heterosexuals.

"gay” for “homosexual” can’t have had a very wide currency in the 1930’s and 40’s.  Disney’s movie “The Three Caballeros” was released in 1945; its theme song began ”We’re three caballeros, three gay caballeros”....I recently saw a reissue of the movie on CD - the words of the song had been changed. The caballeros were something else than gay (don’t remember what - but definitely not gay).  In 1939 a popular song (about heterosexual love) called “South of the Border” had the words ....”for it was fee-yesta, and we were so gay, south of the Border, down Mexico way”..... Perhaps “gay” at that time was more of an “in” word used by homosexuals to each other? In the UK at that time, any homosexual act was still a felony. Don’t know what the situation was in the US - probably different in different States?

Is “queer” politically correct in Australia even when used by a heterosexual? 
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Yes, I think it is fair to say that it has by now been thoroughly reclaimed and reformed.

Perhaps “gay” at that time was more of an “in” word used by homosexuals to each other?

Yes, exactly.