ODO insists on set-up with a hyphen.
OED has this, again under set-up:
a. The way in which something is organized, arranged, or constituted; an organization, arrangement, system, or situation; U.S., personal bearing or carriage; Billiards, etc., a position of the balls (e.g. as left by the last player) from which it is easy to score.
Set-up occurs in a vague and indefinite sense in a large number of contexts, but several reasonably distinct areas of use can be isolated: (a) a business or administrative structure or organization; also, an economic, social, or political system (both with reference to the system or the persons involved); (b) a domestic situation, as determined either by lifestyle or personal relationships; (c) a team (esp. in Sport); (d) the layout of some mechanical apparatus or equipment.
but provides a citation
1953 W. S. Burroughs Junkie 7 "But these people were jerks for the most part and, after an initial period of fascination, I cooled off on the setup."
So yes, setup is a recognised, if somewhat deprecated, form.
The omission of hyphens is not uncommon, though. And as coleopterist has commented, ODO does not insist on the hyphen in American English (the American English sentence was a later addition to the question).