  We establish the (non-lattice) local limit theorem for products of i.i.d.
random variables on an arbitrary simply connected nilpotent Lie group $G$,
where the variables are allowed to be non-centered. Our result also improves on
the known centered case by proving uniformity for two-sided moderate deviations
and allowing measures with a moment of order $2(\dim G)^2$ without further
regularity assumptions. As applications we establish a Ratner-type
equidistribution theorem for unipotent walks on homogeneous spaces and obtain a
new proof of the Choquet-Deny property in our setting.
