  We establish a method for calculating the Poincar\'e series of moduli spaces
constructed as quotients of smooth varieties by suitable non-reductive group
actions; examples of such moduli spaces include moduli spaces of unstable
vector or Higgs bundles on a smooth projective curve, with a Harder-Narasimhan
type of length two. To do so, we first prove a result concerning the smoothness
of fixed point sets for suitable non-reductive group actions on smooth
varieties. This enables us to prove that quotients of smooth varieties by such
non-reductive group actions, which can be constructed using Non-Reductive GIT
via a sequence of blow-ups, have at worst finite quotient singularities. We
conclude the paper by providing explicit formulae for the Poincar\'e series of
these non-reductive GIT quotients.
