Let $\Omega \subset \mathbb{R}^n$ be an open and bounded set and suppose $\partial \Omega$ is smooth on a relatively open subset $\Gamma \subset \partial \Omega$. Consider a weak solution of the Dirichlet problem \begin{align} \Delta u &= f \text{ in } \Omega \\ u &= g \text{ on } \partial \Omega \end{align} with $f$ and $g$ both smooth. If $x_0 \in \Gamma$, is it true that $u$ is smooth up to the boundary in a neighborhood of $x_0$?
I believe this is true since one usually proves boundary regularity by deriving local estimates on the boundary. However, I have never seen the theorem stated on partial boundaries. Can someone confirm this or point me to a reference?
Yes, this is true. As you say, at each $x_0 \in \Gamma$ you find a neighbourhood $U \subset \mathbb R^n$ which can be locally flattened, in that there exists a diffeomorphism $$ \psi : U \to B(0,1) $$ such that $$\psi(U \cap \Omega) = B^+(0,1) := \{ x \in B(0,1) : x_n>0\} \quad\text{ and }\quad\psi(U \cap \partial\Omega) = \{x \in B(0,1) : x_n= 0\}.$$ Then you derive an equation for $u \circ \psi^{-1}$. From this point onwards, you have an elliptic equation purely in $B^+(0,1)$, and local regularity results gives estimates in e.g. $B^+(0,\frac12)$. Some texts will implicitly use these local regularity arguments in their proof, while others like Gilbarg & Trudinger will state the flattened case as separate lemmas.
This result is often assumed implicitly when studying piecewise regular domains, such as polyhedral domains. Grisvard's "Elliptic problems in nonsmooth domains" is a classical reference in this direction, but I don't think it explicitly states this result in the text.
The reason most texts don't state regularity results for domains with regular pieces, is because PDE results in general are rarely stated in the full generality they apply - otherwise there will be far too many technicalities and edge cases to include. What's important are the techniques, and one needs to be able to adapt these to the particular situation one studies - especially for nonlinear problems.