Based on this question, for the Hull-White model of the form
\begin{align*}
dr_t = (\theta(t)-a r_t) dt + \sigma dW_t,
\end{align*}
where $a$ and $\sigma$ are constants, $a(t)$ is a deterministic function, and $W_t$ is a standard Brownian motion, the price at time $t$ of a zero-coupon bond with maturity $T$ and unit face value is given by
\begin{align*}
P(t, T) &= A(t, T) e^{-B(t, T) r_t},
\end{align*}
where
\begin{align*}
B(t, T) = \frac{1}{a}\Big(1-e^{-a(T-t)} \Big),
\end{align*}
and
\begin{align*}
A(t, T) &= \exp\left(- \int_t^T \theta(u) B(u, T) du -\frac{\sigma^2}{2a^2}\big(B(t, T) -T+t\big)-\frac{\sigma^2}{4a}B(t, T)^2\right).
\end{align*}
Note that
\begin{align*}
\ln P(t, T) = \ln A(t, T) -B(t, T) r_t.
\end{align*}
Therefore,
\begin{align*}
d\ln P(t, T) &=\frac{\partial \ln A(t, T)}{\partial t}dt - r_t \frac{\partial B(t, T)}{\partial t} dt- B(t, T) dr_t\\
&=\frac{\partial \ln A(t, T)}{\partial t}dt - r_t \frac{\partial B(t, T)}{\partial t} dt- B(t, T) (\theta(t)-a r_t) dt - \sigma B(t, T) dW_t.
\end{align*}
That is, the zero-coupon bond price volatility is of the form
\begin{align*}
\sigma B(t, T) = \frac{\sigma}{a}\Big(1-e^{-a(T-t)} \Big).
\end{align*}
