What is the minimum pumping length of the union of two languages?
Consider the following two languages:
$$
L_0 = 1(0+1)^* + 0^{n_0}, \\
L_1 = 0(0+1)^* + 1^{n_1}.
$$
The minimum pumping length of $L_0$ is $n_0 + 1$, and that of $L_1$ is $n_1 + 1$. Yet $L_0 \cup L_1 = (0+1)^+$ has a minimum pumping length of 1.

What does hold is that the pumping length of $L_0 \cup L_1$ is at most the maximum of the pumping lengths of $L_0,L_1$. In fact, it can be any number between 1 and that maximum. Indeed, suppose that $1 \leq a,b \leq c$, and consider the languages
$$
\begin{align*}
L_0 &amp;= 1(0+1)^* + 0^{a-1}, \\
L_1 &amp;= 0(0+1)^* + 1^{c-1} + 2^{b-1}.
\end{align*}
$$
The maximum pumping length of $L_0$ is $a$, that of $L_1$ is $c$, and that of $L_0 \cup L_1 = (0+1)^+ + 2^{b-1}$ is $b$.