Here are some references that may help answer your question:
Charles B. Thomas, Nilpotent groups and compact $3$-manifolds, Proc. Cambridge Philos. Soc. 64 (1968), 303-306; MR0233359.
Benny Evans and Louise Moser, Solvable fundamental groups of compact $3$-manifolds, Trans. Amer. Math. Soc. 168 (1972), 189–210; MR0301742.
Peter Teichner, Maximal nilpotent quotients of $3$-manifold groups, Math. Res. Lett. 4 (1997), no. 2-3, 283-293; MR1453060.
A little nugget (due to John Milnor): among the Brieskorn manifolds $\Sigma(p,q,r)$, the only nilmanifolds are $\Sigma(2,3,6)$, $\Sigma(2,4,4)$, and $\Sigma(3,3,3)$, which are circle bundles over the torus with Euler number $1$, $2$, and $3$, respectively.
