  This paper initiates the study of effective twisted conjugacy separability
for finitely generated groups, which measures the complexity of separating
distinct twisted conjugacy classes via finite quotients. The focus is on
nilpotent groups, and our main result shows that there is a polynomial upper
bound for twisted conjugacy separability. That allows us to study regular
conjugacy separability in the case of virtually nilpotent groups, where we
compute a polynomial upper bound as well. As another application, we improve
the work of the second author by giving a precise calculation of conjugacy
separability for finitely generated nilpotent groups of nilpotency class 2.
