I don't know the precise answer (because it depends on the precise definition of an homology theory you may prefer), but I would ask the question in a slightly different way (giving out the CW-structures for a while). We might try to think of the problem in the same way we solve the problem of a topological Galois theory over a space X: we shall get a Galois correspondance
{covers of X}<->{representations of π₁(X)}
iff (the topos of sheaves over) X is locally simply connected. If X is not locally simply connected, we shall get such a correspondance with the toposic π₁(X), which will be a pro-groupoid, hence something different from π₁(X) computed in the usual model category of spaces. We might think of this phenomena as a non additive version of our question.
More generally, given a space X, we might ask when do the topos-theoretic invariants and the CW-invariants (i.e. the ones obtained using the classical model structure on Top) agree. The interest of this point of view is that this will be a local problem on X, and that it will provide systematically two ways to look at a (co)homology theory: in the model category of spaces, or in the homotopy theory of (∞-)topoi.
As far as I understand all this, the only thing we use in the CW-structure is the property of local contractibility: if X is locally contractible, then we can consider the set O(X) of contractible open subspaces of X (ordered by inclusion). We have then that X is the homotopy colimit of its contractible open subspaces (in the usual model category of spaces, but also in the setting of ∞-topoi). Hence the homotopy type of X (as a space and as a topos) is just the nerve of O(X), which has a canonical CW-structure
(but I don't know how to prove that X has a CW-structure itself), so that uniqueness of (co)homology will hold on such an X (assuming some nice descent properties). The idea is that we can weaken these local conditions.
To set the general problem, we might fix a nice left Bousfield localization of the model category of topological spaces, which I shall denote by S. By nice left Bousfield localization, we might mean that it is obtained by a nullification (i.e. by inverting a map of shape A->pt). We might think of S as the homotopy theory of (n-1)-groupoids (in the case A=Sⁿ, for 0≤n≤∞), but we might also take A such that S corresponds to CW-complexes up to Quillen's plus construction, which is more related with our problem with singular homology. The reasons why I suggest to consider only nullifications are that 1) they are nicer because they lead to proper model categories (at least in Top), which makes computations easier, 2) in all the definitions, we shall only deal with weak equivalences of shape X->pt. The idea is to study the theory of "topoi relatively to S".
Given an ∞-topos X, let us denote by Sh(X) the ∞-category of stacks on X with values in S. We thus have Sh(pt)=S. Let us say that X is S-aspherical (or, if you prefer, S-acyclic...) if the constant stack functor S->Sh(X) is fully faithful. And let us say that X is locally S-aspherical if there exists a generating family in X (if X is a topological space, this means a basis of open subspaces) made of stacks U over X such that X/U is aspherical. For a general X, the constant stack functor c:S->Sh(X) will always have a pro-adjoint, and the nice thing is that this pro-adjoint will be a genuine left adjoint to c if and only if X is locally S-aspherical. In other words, we then have a functor à la Artin-Mazur
{∞-topoi}->Pro(S)
(which has a right adjoint; see 7.1.6.15 in Lurie's book on higher topoi),
and I guess it will induce an equivalence of ∞-categories of shape
{locally S-aspherical ∞-topoi}=S
(for this, consider {locally S-aspherical ∞-topoi} as localized by S-equivalences, i.e. by maps which become invertible in Pro(S)).
Thus, for a topologcal space X, we have two ways to send it to Pro(S): consider the pro-S-homotopy type associated to Sh(X), or consider X as an object of the categoy underlying the model category structure on Top underlying the definition of S.
Then, topological spaces X whose associated ∞-topos are locally S-aspherical are precisely the one for which the two ways to see them in Pro(S) coincide. If S is the homotopy theory associated to Quillen's plus construction, then we get that, for any topological space X which has a basis of open subspaces U such that U->pt induces an isomorphism in (pro)homology with constant coefficients, the topos theoretic singular (co)homology and the "usual" singular (co)homology (computed using Hom's in (pro-)spectra) will coincide. But otherwise, they won't, and we might expect to get explicit examples of non-agreement in this way.
[additional comments and precisions]
What I call the topos theoretic singular cohomology is just the sheaf cohomology with coefficients in the constant sheaf Z (which is the same as taking the global sections of the image of the K(Z,n)'s by c:S->Sh(X)). If you consider a topological space X, then the topos-theoretic singular cohomology of X will be a cohomology theory satisfying excision, which will coincide with usual singular cohomology for locally contractible spaces (and more generaly, locally S-aspherical spaces, where S denotes the homotopy theory of spaces up to Quillen's plus construction). For homology, you will have the same picture, except that it will take values in pro-abelian groups in general (except again for locally contractible spaces, etc): consider the associated pro-space, and take its usual homology. This toposic point of view thus gives you a cohomology theory which coincides with singular (co)homology for good spaces, but won't agree for a large class of them. All this story applies for any (co)homology theory which preserves homotopy colimits (which implies the Eilenberg-Steenrod axioms).
As for the existence of the pro-adjoint of c:S->Sh(X), this is because any accessible left exact functor has such a thing (as proved in Lurie's book). The pro-homotopy type associated to X is then the image of the terminal object of Sh(X) by this pro-adjoint.
I should mention as well that the theory of pro-homotopy types of topoi is treated in section 7.1.6 of Lurie's book on infinity topoi (in the case where S is the usual homotopy theory of infinity-groupoids, but the same arguments should work in the generality I suggested here). I cannot resist to mention that this kind of point of view can be seen (after Lurie) as a conceptual way to look at shape theory, but this is also (higher) Galois theory in the sense of Grothendieck (such a point of view is also mentionned in the work of Toën and Vezzosi on higher topoi): in the case where S is the homotopy theory of 1-groupoids, one gets exactly the theory of the fundamental group as developped in SGA1 (and later by Leroy, Moerdijk, etc).
[Conjecture]
I propose the following conjecture as an answer to the initial question (and the constructions suggested above give at least some evidence, not to say an idea for a strategy of proof).
The largest class of spaces for which the Eilenberg-Steenrod axioms determine singular homology consists of the topological spaces X satisfying one of the following four equivalent conditions.
1) for any open subspace U of X, sheaf-theoretic homology and usual singular homology coincide;
2) the open subspaces U of X such that the sheaf theoretic singular homology of U is trivial in degree >0 form a basis of open subsets of X;
3) the open subspaces U of X such that the usual singular homology of U is trivial in degree >0 form a basis of open subsets of X;
4) the open subspaces U of X such that U_+ is contractible form a basis of open subsets of X.
There are maps $|Sing(X)| \to |\underline{Sing}(X)| \to X$ which realize to weak homotopy equivalences. The inclusion of the n-skeleton $|Sing(X)|^{(n)}| \to |Sing(X)|$ is n-connected, because this is always true for CW-complexes, and so the map $|Sing(X)|^{(n)} \to X$ is n-connected. You can't really do any better than this estimate because the n-skeleton has zero homology groups in degrees above n.
The simplicial space $\underline{Sing}(X)$ contains the sub-simplicial space of constant simplices $\Delta^n \to X$. This is homeomorphic to $X$ itself and so, if we write $cX$ for the constant simplicial space with value $X$, we get a map $cX \to \underline{Sing}(X)$. This inclusion $X \to \underline{map}(\Delta^n,X)$ is a homotopy equivalence because the simplex is contractible, so this map of simplicial spaces is levelwise a weak equivalence. The geometric realization of $cX$ is $X$ itself, and so is its n-skeleton for all n. An excision argument (which takes some work) will show that the same is true for the simplicial space (at least under good conditions), and so each of the skeleta of $|\underline{Sing}(X)|$ is homotopy equivalent to $X$.
Therefore the map on n-skeleta is n-connected. I realize that this is the "compare with $X$" game that you mentioned, but my point is that because the simplicial space $\underline{Sing}(X)$ is homotopically constant the comparison $|Sing(X)| \to |\underline{Sing}(X)|$ really is comparing with $X$. From the point of view of homotopy theory it's not arising from good levelwise structure of the map at all, but comes from the simplicial assemblage.
I don't know whether this helps your generalization.
Best Answer
This answer is simply to write the details for my comment above. It amounts to doing a little more work with homotopy equivalences, so as to carry out essentially the argument you gave in your comment regarding the smooth case.
Assume that $f:X\to Y$ is a weak equivalence of topological spaces. We want to show that the map induced on topological bordism (as you define it) is a bijection.
Regarding the surjectivity, I will simply repeat the content of Tom Goodwillie's comment. First, we observe that any topological manifold has the homotopy type of a CW-complex (you can find references to proofs by Hanner in Milnor's "On spaces having the homotopy type of a CW-complex", as remarked in the comment of Greg Friedman). It follows that $f$ induces a bijection on homotopy classes of maps $f_\ast :[M,X]\to [M,Y]$. Therefore, any map $g:M\to Y$ is homotopic and thus cobordant to a map which lifts to $X$. This proves surjectivity.
To prove injectivity, we need to know that the inclusion of the boundary of a topological manifold is a closed Hurewicz cofibration, that is a NDR-pair. This holds since the boundary of a topological manifold is collared (see chapter 2 of Ferry's notes). With this fact at hand, we can show that for any topological manifold $M$, the pair $(M,\partial M)$ is homotopy equivalent rel $\partial M$ to a pair $(A,\partial M)$ such that the inclusion $\partial M\to A$ is a Serre cofibration.
Proof of claim: Simply factor the inclusion $\partial M\to M$ as a Serre cofibration followed by a weak equivalence $\partial M\to A\to M$. Since $\partial M$ is a topological manifold, it has the homotopy type of a CW-complex, and it follows that $A$ also has the homotopy type of a CW-complex. Then the map $A\to M$ is a weak equivalence of spaces with the homotopy type of CW-complexes, and thus a homotopy equivalence. The map $A\to M$ is therefore a homotopy equivalence rel $\partial M$, since both maps $\partial M\to A$ and $\partial M \to M$ are Hurewicz cofibrations.
We can now prove injectivity by carrying out an argument similar to the one you wrote in your comment. Given bordism classes $g_0:M_0\to X$, $g_1:M_1\to X$ in $X$ and a cobordism $g:M\to Y$ ($\partial M=M_0\coprod M_1$) between $f\circ g_0$ and $f\circ g_1$, we need to find a cobordism $g':M\to X$ between $g_0$ and $g_1$. For that purpose, we replace $(M,\partial M)$ with $(A,\partial M)$ as described above. Since $\partial M\to A$ is a Serre cofibration, we can find a lift up to homotopy $A\to X$ of the composite $A\to M\to Y$, which furthermore extends the map $(g_0,g_1):\partial M\to X$. Composing with the homotopy equivalence $M\to A$ rel $\partial M$, we get the desired map $g':M\to X$.