For two-step nilpotent groups, residual finiteness growth is at most log^{d+1}, with d an invariant of the complex Mal'cev completion, and is exactly log^{d+1} when the commutator subgroup has rank at most 2.
Short laws for finite groups and residual finiteness growth
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Residual Finiteness Growth in Two-Step Nilpotent Groups
For two-step nilpotent groups, residual finiteness growth is at most log^{d+1}, with d an invariant of the complex Mal'cev completion, and is exactly log^{d+1} when the commutator subgroup has rank at most 2.