Is there a treatment in the literature of the structure sets relating simple homotopy equivalences to homeomorphisms in the three dimensional case? I am aware that due to the geometrization conjecture, homotopy equivalences and homeomorphism agree. I am also aware of Jahren and kwasik's paper Threedimensional surgery theory, UNilgroups and the Borel conjecture in Topology 42 (2003) 1353–1369.

1$\begingroup$ I am not sure I understand your first claim: there are lens spaces which are homotopy equivalent but not homeomorphic. If you have no lens spaces in your prime decomposition, then okay, you're fine. But it is a result of Turaev (assuming geometrization) that simple homotopy type and homeomorphism type of 3manifolds coincide. $\endgroup$– mmeNov 13 '18 at 14:46
It is well known that the surgery sequence completely breaks down in dimension 3, so looking at structure sets is not really the right thing to do. For instance, in the classification of highdimensional lens spaces, a given simple homotopy type (as in Mike Miller's comment) will contain many homeomorphism types (detected by `multisignatures'). But geometrization says that this doesn't hold in dimension 3. This represents a failure of the Lgroup $L_4$ to act on the structure set of (say) a lens space.
Likewise, there are normal maps whose corresponding surgery problems with finite fundamental group have trivial surgery obstructions, but don't have a solution. Again this is from geometrization, which tells you that the finite fundamental groups of 3manifolds are those of spherical space forms. You can learn about this in the book A survey of the spherical space form problem by Davis and Milgram.
Such failures are traceable to either of these facts: (1) surgery along circles in a 3manifold generally changes the fundamental group, or (2) it's not so easy to find embedded spheres in 4manifolds. The best one can usually do is surgery up to homology equivalence. This is basically folklore; it is briefly discussed in the KirbyTaylor article "A survey of 4manifolds through the eyes of surgery".
There are some consistencies with surgery theory and 3manifold theory; for instance I seem to recall that (with a few exceptions) the fundamental groups of Haken manifolds have vanishing $L_3$, which is consistent with Waldhausen's theorem that such manifolds are classified by their fundamental group.
Edit: There is more about what surgery theory looks like in dimension 3 in an article of JahrenKwasik (Threedimensional surgery theory, UNilgroups and the Borel conjecture, Topology 42 (2003) 1353–1369)