Ru -- the Leray spectral sequence exists for any map $f:X\to Y$ of arbitrary topological spaces and any sheaf $F$ on $X$ and its second term is $$E_2^{p,q}=H^p(Y,R^q f_*F)$$ where $R^q f_*F$ are the sheaves on $Y$ that are obtained by sheafifying the presheaves $U\mapsto H^q(f^{-1}(U),F)$. Here are some remarks that might help:
If $f$ is a locally trivial fibration and $F$ is constant then all $R^q f_*F$ are locally constant; if in addition $Y$ is simply-connected then the sheaves are constant and we can express $E_2$ in terms of the constant cohomology of $Y$.
It may happen that all fibers $f^{-1}(y),y\in Y$ are homeomorphic but some or all $R^q f_*F$ are non-constant; take e.g. $X=(\mathbb{R}\setminus \{ 0\})\sqcup \{ 0\}, Y=\mathbb{R},f$ the identity map.
Nevertheless, if $f:X\to X/G$ where $G$ is a connected Lie group that acts nicely on $X$ (say so that the quotient is Hausdorff) with finite stabilizers, and $F$ is a constant sheaf with stalk $\mathbb{Q}$ (or $\mathbb{R}$ or $\mathbb{C}$) then any sheaf $R^q f_*F$ is constant with stalk $H^q(G,\mathbb{Q})$ (resp., $H^q(G,\mathbb{R})$ and $H^q(G,\mathbb{C})$).
Two possible references (which means, to be honest, that there may be better references but that's where I first learned this from) are Godement, Topologie alg\'ebrique et th\'eorie des faisceaux, the very end of chapter 4, and Griffiths-Harris, the very end of vol.1