Suppose a CW complex $M$ is given by the union of $n$-spheres, namely $M=\bigcup_{\alpha\in A}S^n$, without knowledge of intersections. The only requirement is that the union covers $M$. Let $\Sigma=\{S^n,\dots,S^n\}$ be a finite collection of sets, with cardinality $|A|$. The nerve consists of all subcollections
whose sets have a non-empty common intersection, $\text{Nrv}(\Sigma)=\left\{X\subseteq\Sigma\big|\bigcap X\ne\emptyset\right\}$, which is an abstract simplicial complex. The nerve should look something like this (e.g. a Čech complex):
That is, we are allowed to arrange the spheres in a configuration of our choosing, so long as the configuration still covers $M$. (Indeed, we can "pull" the spheres apart as much as possible so that they still cover $M$-an optimal configuration-with the least amount of spheres used).
How do we compute homology groups $H_k(M;\mathbb{Z})$ from the nerve $\text{Nrv}(\Sigma)$ for $\Sigma=\{S^n,\dots,S^n\}$ covering the CW complex $M$?
Idea: Since we need some information on intersections, suppose we construct the following optimal configuration. Begin with two $n$-spheres attached at a base point, namely $S^n\vee S^n$. Then construct two other $n$-spheres that pass through the intersection point of $S^n\vee S^n$. Finally, we continue the process by constructing other spheres $S^n$ that pass through the intersection "points" of other spheres. We write "points" realizing that the intersection of two $n$-spheres is actually an $(n-1)$-sphere. The points we refer to are those corresponding to the two intersections in the $S^2$ orthographic projection onto a plane. We can of course vary the radius of the sphere under this construction.
As mentioned by Mike Miller, the condition we want is that the $k$-fold intersections, for $k$ sufficiently large, are all empty or contractible. Then the Čech complex of this cover (with constant sheaf $\mathbb{Z}$) recovers the homology of the manifold. The general case where $k$-fold intersections are not contractible instead takes the form of a spectral sequence involving cohomology of the various intersections. I am not sure, however, how to make this mathematically precise.
Any help would be much appreciated. Thanks in advance!