... of CW-complexes is homotopy equivalent to a Serre cofibration, and thus must be a mixed cofibration. In particular, for any topological manifold $M$, the inclusion $\partial M\to M$ is a mixed cofibration. Using this, the last paragraph of the answer can be replaced by a single application of the lifting axioms in the mixed model structure. That is, after factoring $f:X\to Y$ as a trivial Strom cofibration (a strong deformation retract) followed by a Hurewicz fibration --- this factorization actually coincides with the one in the mixed model structure.

You are correct. I don't know much about mixed model structures, other than what is at the nlab page (ncatlab.org/nlab/show/mixed+model+structure). Nevertheless, a simple relative homotopy lifting argument for the Hurewicz/Strom model structure shows that any Strom cofibration which is homotopy equivalent to a Serre cofibration is also a mixed cofibration (i.e. has the left lifting property with respect to Hurewicz fibrations which are weak equivalences). The argument in the penultimate paragraph of the answer shows any Strom cofibration between spaces of the homotopy type...

@Justin: No, but we can prove that it is homotopy equivalent to a Serre cofibration. I have tried to write some details in an answer below.

@Karol: In the topological case, it is an old theorem that the boundary of a manifold is collared. I think it is due to Brown in "Locally flat embeddings of topological manifolds". You can find the proof there or in chapter 2 of Ferry's notes at math.rutgers.edu/~sferry/ps/geotop.pdf.

Adding slightly to Tom's comment, the invariance under weak equivalence of topological bordism (as you define it) should follow from the following two facts: (1) any topological manifold has the homotopy type of a CW-complex, and (2) the inclusion of the boundary of a topological manifold is a cofibration (since it is collared).

@Tom: I made a mistake. I hope I got it right now.

@BS: Thanks. I added a generation condition for the map $M\to N$.