I have skipped some details below for brevity.
First, notice that $C$ is contained in a half-space that separates the set from $\tau$, specifically $\langle w-\tau,\tau'-\tau \rangle \geq ||\tau-\tau'||^2$. This is the first order optimality condition for the distance minimization problem.
(To see this, note that $|| (\lambda w + (1-\lambda) \tau')-\tau||^2 - ||\tau'-\tau||^2 \geq 0$, $\forall \lambda \in [0,1]$, by convexity. Now expand the expression, divide across by $\lambda>0$ and then let $\lambda \rightarrow 0$. )
Then form the following estimate: $|| w - \tau||^2 = ||w - \tau' + \tau' - \tau||^2 = || w - \tau'||^2 + 2 \langle w-\tau',\tau'-\tau \rangle + ||\tau' - \tau||^2$
$= || w - \tau'||^2 + 2 (\langle w-\tau,\tau'-\tau \rangle -||\tau' - \tau||^2)+ ||\tau' - \tau||^2$
$\geq || w - \tau'||^2 + ||\tau' - \tau||^2$. This is slightly stronger than the result you were looking for.