  In Part I, we defined a large class of open symplectic manifolds, called
symplectic $\mathbb{C}^*$-manifolds, which are typically non-exact at infinity,
and we showed that their Hamiltonian Floer theory is well-defined. In this
paper, we construct Morse-Bott-Floer spectral sequences for these spaces,
converging to their symplectic cohomology. These are a Floer-theoretic analogue
of the classical Morse-Bott spectral sequence for ordinary cohomology. The
spectral sequences determine a filtration by ideals on quantum cohomology. We
compute a plethora of explicit examples, each highlighting various features,
for Springer resolutions, ADE resolutions, and several Slodowy varieties of
type A. We also consider certain Higgs moduli spaces, for which we compare our
filtration with the well-known $P=W$ filtration. We include a substantial
appendix on Morse-Bott-Floer theory, where a large part of the technical
difficulties are dealt with.
