Treewidth and the NL vs L Problem
It appears the problem is in L by [EJT10] and thus L-complete under $\text{NC}^1$ reducibility by [CM87]. See page 2 of [EJT10]:


  Applying Theorem I.3 to the formula $\phi(X)$ expressing that $X$ is a
  simple path from $s$ to $t$ shows that the problem $\{(G,s,t) \text{ }
  | \text{ tw$(G) \leq k$, there is a path from $s$ to $t$ in $G$}\}$
  lies in L


Actually this result applies to all problems on bounded treewidth graphs which can be formulated in monadic second-order logic in L.

[EJT10] Michael Elberfeld, Andreas Jakoby, and Till Tantau. Logspace versions of the theorems of Bodlaender and Courcelle. In Proceedings of the 51st Annual Symposium on Foundations of Computer Science (FOCS), pages 143–152, 2010.

[CM87] Stephen A. Cook, Pierre McKenzie: Problems Complete for Deterministic Logarithmic Space. J. Algorithms 8(3): 385-394 (1987)