The two coordinate systems $(x,y)$ and $(\tau,n)$ are both orthogonal, which is the only important thing.
So there exists an (orthogonal) rotation transformation
$$
\begin{cases}
\tau = \cos(\alpha)\,x-\sin(\alpha)\,y \\
n = \sin(\alpha)\,x+\cos(\alpha)\,y
\end{cases}
$$
We want to know how the derivatives of the solution $\,u\,$ transform.
$$
\frac{\partial u}{\partial x} =
\frac{\partial u}{\partial \tau}\frac{\partial \tau}{\partial x}
+ \frac{\partial u}{\partial n}\frac{\partial n}{\partial x} \\
\frac{\partial u}{\partial y} =
\frac{\partial u}{\partial \tau}\frac{\partial \tau}{\partial y}
+ \frac{\partial u}{\partial n}\frac{\partial n}{\partial y}
$$
It follows, with operator notation:
$$
\frac{\partial}{\partial x} u =
\left[ \cos(\alpha)\frac{\partial}{\partial \tau}
+\sin(\alpha)\frac{\partial}{\partial n} \right] u\\
\frac{\partial}{\partial y} u =
\left[ -\sin(\alpha)\frac{\partial}{\partial \tau}
+\cos(\alpha)\frac{\partial}{\partial n} \right] u
$$
So we have:
$$
\frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} =
\left(\frac{\partial}{\partial x}\right)^2 + \left(\frac{\partial}{\partial y}\right)^2 = \\
\left[\cos(\alpha)\frac{\partial}{\partial \tau} +\sin(\alpha)\frac{\partial}{\partial n} \right]^2
+ \left[ -\sin(\alpha)\frac{\partial}{\partial \tau}+\cos(\alpha)\frac{\partial}{\partial n} \right]^2 = \\
\left[\cos^2(\alpha)\left(\frac{\partial}{\partial \tau}\right)^2 +
2\sin(\alpha)\cos(\alpha)\frac{\partial}{\partial \tau}\frac{\partial}{\partial n}
+\sin^2(\alpha)\left(\frac{\partial}{\partial n}\right)^2\right] + \\
\left[\sin^2(\alpha)\left(\frac{\partial}{\partial \tau}\right)^2
-2\sin(\alpha)\cos(\alpha)\frac{\partial}{\partial \tau}\frac{\partial}{\partial n}
+\cos^2(\alpha)\left(\frac{\partial}{\partial n}\right)^2\right] = \\
\left(\frac{\partial}{\partial \tau}\right)^2 + \left(\frac{\partial}{\partial n}\right)^2 =
\frac{\partial^2}{\partial \tau^2} + \frac{\partial^2}{\partial n^2}
$$
Therefore $\Delta u$ is the same in the (global) plane coordinates as well as in the (local) curve coordinates.