The result answering this question is the following: Suppose (M,F) is a foliation, with holonomy groupoid G = Hol(M,F). Let p: M --> G be the "quotient map from M to the leafspace" , with associated pullback p* from Sheaves(G) to sheaves on M. Suppose the holonomy cover of each leaf is n-connected. Then for any abelian sheaf A on M equipped with an action of the holonomy groupoid, i.e. for any object of Sheaves(G), the map p* induces isomorphisms in cohomology
H*(G,A) --> H*(M,p*A) in degrees 0,...,n.

The proof is in fact not too hard, and surely much easier than what I did for the classifying space. It is a combination of some generalities for etale groupoids, and a similar fact for submersions between manifolds: If f: N --> M is a submersion with n-connected fibers, the f induces isomorphisms in cohomology in degrees up to n, for coefficients in any sheaf on M.

Ieke Moerdijk