What (if any) is the relationship between the conserved (non-topological) noetherian charges and topological charges? Namely, is there any "generalization" of the Noether's first theorem that includes topological charges as subcase and it provides some relationship between these two classes of charge?
Let there be given a physical system. What charges are Noetherian and what charges are topological often depend on the precise action formulation and field content of the physical system, see also this Phys.SE post. To simplify the discussion, let us assume that the action formulation is fixed, and below definitions will then refer to this fixed formulation.
A characteristic feature of a topological conservation law (CL) is that it holds off-shell, i.e. independent of the Euler-Lagrange equations. Olver calls such a CL trivial of the second kind, cf. Ref. 1.
Many authors require by definition a topological CL to not be a Noetherian CL, i.e. not covered by a Noetherian theorem.
Nevertheless, even if there exists a quasi-symmetry (QS) of the action that via Noether theorem leads to a topological CL, the CL is (as mentioned above) trivial, which (under mild assumptions) means that it corresponds to a trivial QS, cf. Ref. 1. (A trivial QS means that the QS transformation vanishes on-shell. See also this Phys.SE post.)