One of the main features of the Hull-White model is that it matches the market at $t = 0$.
This means that at $t = 0$, not only does the zero coupon bond prices (starting from zero) not depend on the volatility, but neither do they depend on the mean reversion level. These prices depend only on the zero curve observed in the market.
Of course, this is not to be confused with future zero bond prices $P(t,T)$ seen from $t = 0$, which are random variables and as a result have a distribution depending on the volatility and mean reversion as well.
To show that the ZC bond prices at $t = 0$ match the market and don't depend on the model parameters, I will use a different (more convenient) formulation (see e.g. Andersen and Piterbarg, section 10.1.2.2), which uses $x(t) = r(t) - f(0,t)$ instead of the short rate $r(t)$. Which leads to the following SDE (keeping your notations):
$$
\begin{aligned}
x(0) &= 0 \\
dx(t) &= \left( y(t) - a x(t) \right) dt + \sigma dW(t)
\end{aligned}
$$
with: $y(t) = \frac{\sigma^2}{2a} \left(1-e^{-2at} \right)$.
The ZC bond price is given by:
$$
P(t,T) = \frac{P^M(0,T)}{P^M(0,t)}\exp \left(-\frac{1}{2}B(t,T)^2y(t) - B(t,T)x(t) \right)
$$
In the formula above, I used the superscript $^M$ to denote that the $P^M$ prices come from the zero curve observed in the market.
Taking $t = 0$, as $x(0) = y(0) = 0$ and $P^M(0, 0) = 1$, we have:
$$P(t,T) = P^M(0,T)$$