Here is a completely elementary example which shows that group cohomology is not empty verbiage, but can solve a problem ("parametrization of rational circle") whose statement has nothing to do with cohomology.
Suppose you somehow know that for a finite Galois extension $k\subset K$ with group $G$ the first cohomology group $H^1(G,K^*)$ is zero : this is the homological version of Hilbert's Theorem 90 ( you can look it up in Weibel's book on homological algebra, pages 175-176).
If moreover $G $ is cyclic with generator $s$, this implies that an element of $K$
has norm one if and only if it can be written $\frac{a}{s(a)}$ for some $a\in K$.
Consider now the quadratic extension $k=\mathbb Q \subset K=\mathbb Q(i)$ with generator $s$ of $Gal(\mathbb Q (i)/\mathbb Q)$ the complex conjugation.The statement above says that $x+iy\in \mathbb Q(i)$ satisfies $x^2+y^2=1$ iff
$x+iy=\frac{u+iv}{s(u+iv)}=\frac{u+iv}{u-iv}=\frac{u^2-v^2}{u^2+v^2}+i\frac{2uv}{u^2+v^2}$ for some $u+iv\in \mathbb Q (i)$ .
So we have obtained from group cohomology the well-known parametrization for the rational points of the unit circle $x^2+y^2=1$
$$x=\frac{u^2-v^2}{u^2+v^2}, \quad y=\frac{2uv}{u^2+v^2}$$.
Short answer: you don't want to consider group cohomology as defined for finite groups for Lie groups like $U(1)$, or indeed topological groups in general. There are other cohomology theories (not Stasheff's) that are the 'right' cohomology groups, in that there are the right isomorphisms in low dimensions with various other things.
Long answer:
Group cohology, as one comes across it in e.g. Ken Brown's book (or see these notes), is all about discrete groups. The definition of group cocycles in $H^n(G,A)$, for $A$ and abelian group, can be seen to be the same as maps of simplicial sets $N\mathbf{B}G \to \mathbf{K}(A,n)$, where $\mathbf{B}G$ is the groupoid with one object and arrow set $G$, $N$ denotes its nerve and $\mathbf{K}(A,n)$ is the simplicial set corresponding (under the Dold-Kan correspondence) to the chain complex $\ldots \to 1 \to A \to 1 \to \ldots$, where $A$ is in position $n$, and all other groups are trivial. Coboundaries between cocycles are just homotopies between maps of simplicial spaces.
The relation between $N\mathbf{B}G$ and $K(G,1)$ is that the latter is the geometric realisation of the former, and the geometric realisation of $\mathbf{K}(A,n)$ is an Eilenberg-MacLane space $K(A,n)$, which represents ordinary cohomology ($H^n(X,A) \simeq [X,K(A,n)]$, where $[-,-]$ denotes homotopy classes of maps). This boils down to the fact that simplicial sets and topological spaces encode the same homotopical information. It helps that $N\mathbf{B}G$ is a Kan complex, and so the naive homotopy classes are the right homotopy classes, and so we have
$$sSet(N\mathbf{B}G,\mathbf{K}(A,n))/homotopy \simeq Top(BG,K(A,n))/homotopy = [BG,K(A,n)]$$
In fact this isomorphism is a homotopy equivalence of the full hom-spaces, not just up to homotopy.
If we write down the same definition of cocycles with a topological group $G$, then this gives the 'wrong' cohomology. In particular, we should have the interpretation of $H^2(G,A)$ as isomorphic to the set of (equivalence classes of) extensions of $G$ by $A$, as for discrete groups. However, we only get semi-direct products of topological groups this way, whereas there are extensions of topological groups which are not semi-direct products - they are non-trivial principal bundles as well as being group extensions. Consider for example $\mathbb{Z}/2 \to SU(2) \to SO(3)$.
The reason for this is that when dealing with maps between simplicial spaces, as $N\mathbf{B}G$ and $\mathbf{K}(A,n)$ become when dealing with topological groups, it is not enough to just consider maps of simplicial spaces; one must localise the category of simplicial spaces, that is add formal inverses of certain maps. This is because ordinary maps of simplicial spaces are not enough to calculate the space of maps as before. We still have $BG$ as the geometric realisation of the nerve of $\mathbf{B}G$, and so one definition of the cohomology of the topological group $G$ with values in the discrete group $A$ is to consider the ordinary cohomology $H^n(BG,A) = [BG,K(A,n)]$.
However, if $A$ is also a non-discrete topological group, this is not really enough, because to define cohomology of a space with values in a non-discrete group, you should be looking at sheaf cohomology, where the values are taken in the sheaf of groups associated to $A$. For discrete groups $A$ this gives the same result as cohomology defined in the 'usual way' (say by using Eilenberg-MacLane spaces).
So the story is a little more complicated than you supposed. The 'proper ' way to define cohomology for topological groups, with values in an abelian topological group (at least with some mild niceness assumptions on our groups) was given by Segal in
G Segal, Cohomology of topological groups, in: "Symposia Mathematica, Vol. IV (INDAM, Rome, 1968/69)", Academic Press (1970) 377–387
and later rediscovered by Brylinski (it is difficult to find a copy of Segal's article) in the context of Lie groups in this article.
Best Answer
You're only finding UCT in the literature for trivial group actions, because there is no general UCT for nontrivial group actions:
The general Kunneth formula does not hold for arbitrary groups and actions. But it does hold a good amount of times, and I elaborated on this here: Kuenneth-formula for group cohomology with nontrivial action on the coefficient.
Now the UCT, which relates $H_*(G,M)$ to $H_*(G,\mathbb{Z})$, only follows from the Kunneth formula for trivial group actions. The Kunneth theorem considers the tensor product $C_*\otimes D_*$ of two chain complexes, and the special case for UCT is $D_*=M$ for some $R$-module. To guarantee that the images of the boundary maps are $R$-projective, some extra assumptions are needed (like $R$ is a PID). For group homology, we work with $F_*\otimes_{\mathbb{Z}G}M$ where $F_*$ is a free resolution of $\mathbb{Z}$ as a $\mathbb{Z}G$-module. We cannot take $R=\mathbb{Z}G$ (otherwise the assumption about the boundaries would imply that all homology groups are trivial), so we take $R=\mathbb{Z}$. But then $M$ must be trivial as a $\mathbb{Z}G$-module in order to express $F_*\otimes_{\mathbb{Z}G}M$ as $C_*\otimes_\mathbb{Z}M$. In this case, $F_*\otimes_{\mathbb{Z}G}M=(F_*\otimes_{\mathbb{Z}G}\mathbb{Z})\otimes_\mathbb{Z}M$ and we can apply the UCT.
Morally, in lieu of Qiaochu's remark you must ask: Given any data involving the $G$-action, how would the operators $\oplus,\otimes,\text{Tor},\text{Ext}$ encode such information? And as shown above, you can't use that information to pass from $M$ to $\mathbb{Z}$. For example, let $\mathbb{Z}_2$ act on $M=\mathbb{Z}_2\oplus\mathbb{Z}_2$ by swapping the generators of the summands. Where would this maneuver exist on the coefficient $\mathbb{Z}$ or on any homological object? We can't simply "forget" the action, because $H^1(\mathbb{Z}_2,M_\text{nontriv})=0$ while $H^1(\mathbb{Z}_2,M_\text{triv})=M$.