= Solution
We use the <abacus divisible-hook correspondence> in its multiset form: the <hook lengths> of $\lambda$ that are divisible by $p$, divided by $p$, form exactly the disjoint union of the <multisets> of <hook lengths> of the components of its <quotient of a partition> for modulus $p$.
Iterating this correspondence gives, for every $r\geq1$,
$$
\{h/p^r:h\in\mathcal H(\lambda),\ p^r\mid h\}
=\bigsqcup_{\eta\in T^Q(\lambda)_r}\mathcal H(\eta)
$$
as <multisets>. Each <Young diagram> has one hook for every cell, so the <cardinality> of the right-hand side is $|T^Q(\lambda)_r|$, the sum of the component sizes.
For any positive integer $h$, its <P-adic valuation> is the number of positive $r$ for which $p^r\mid h$. Summing this identity over hooks and interchanging the finite sums proves
$$
\begin{aligned}
\nu_p\!\left(\prod_{h\in\mathcal H(\lambda)}h\right)
&=\sum_{h\in\mathcal H(\lambda)}\sum_{r\geq1}\mathbf1_{p^r\mid h}\\
&=\sum_{r\geq1}|T^Q(\lambda)_r|.
\end{aligned}
$$
Thus
$$
\boxed{\nu_p\!\left(\prod_{h\in\mathcal H(\lambda)}h\right)=\sum_{r\geq1}|T^Q(\lambda)_r|.}
$$
The sums are finite because hook lengths are bounded by $|\lambda|$, and the total size at each level of the <quotient tower of a partition> decreases by at least a factor $p$ until it reaches zero.
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