Is it “Unions' Assassins' Guild” or “Union's Assassins' Guild”?
If my English serves me correctly, I think both are right but have slightly different meanings. Can someone shed some light?
The only difference is that in the first, Unions' has the apostrophe (') after the s (Unions'), and in the second, the apostrophe comes before the s (Union's). The difference in meaning is that when the apostrophe comes after the s, it signifies a plural possessive (more than one Union), whereas when it comes before the s, it signifies the singular (one Union). Presumably the Guild's name refers to only one Union, so the second would be more appropriate.
Edit: I must point out an oversight of mine - as per FumbleFinger's post, Union should probably not have an apostrophe or s at all. Also, Assassins doesn't need to be possessive.
Lots of Student Unions have Assassins Guilds, and if we suppose that at least some of those students know or are being taught basic English, we should be able to assume they know how to punctuate the names of their own organisations.
In almost every single instance directly connected to an Assassins Guild, it's just like that (no apostrophes). One exception I noticed was City University London Assassins' Guild - but City doesn't have much of a reputation imho, and I see they say that page will be "seen by 1000's of students", so they obviously like apostrophes (I and most others would simply use 1000s there).
In principle you could have an apostrophe after Unions, or apostrophe+s after Union to indicate "owned by/associated with", but it's grammatically valid (and far more common) to use Union as a noun functioning as adjective, so it just identifies the particular [type/instance of] Assassins Guild being referenced.
Thus even if OP is writing of several such guilds, he can simply call them Union Assassins Guilds - there's no need to pluralise or apostrophise Union, and imho it would be clumsy/stilted to do so.