  We construct several quantum coset W-algebras, e.g. sl(2,R)/U(1) and
sl(2,R)+sl(2,R) / sl(2,R), and argue that they are finitely nonfreely
generated. Furthermore, we discuss in detail their role as unifying W-algebras
of Casimir W-algebras. We show that it is possible to give coset realizations
of various types of unifying W-algebras, e.g. the diagonal cosets based on the
symplectic Lie algebras sp(2n) realize the unifying W-algebras which have
previously been introduced as `WD_{-n}'. In addition, minimal models of WD_{-n}
are studied. The coset realizations provide a generalization of
level-rank-duality of dual coset pairs. As further examples of finitely
nonfreely generated quantum W-algebras we discuss orbifolding of W-algebras
which on the quantum level has different properties than in the classical case.
We demonstrate in some examples that the classical limit according to Bowcock
and Watts of these nonfreely finitely generated quantum W-algebras probably
yields infinitely nonfreely generated classical W-algebras.
