My new paper with Carlos Florentino and Jaime Silva answers this question:
Mixed Hodge structures on character varieties of nilpotent groups.
In particular, see Section 4.4.
For an implementation of the MHS in some special cases, please see the Mathematica NB here:
https://github.com/seanlawton/Mixed-Hodge-structures-on-character-varieties-of-nilpotent-groups