The easiest way I think to get what you want is to use Omar Antolín-Camarena's approach.
I just want to point out that the result is actually a very special case of something that is much more generally true: *in certain special cases, one can commute a homotopy limit with a homotopy colimit.* (See below).

Suppose $I \to \text{Top}/X$ is a diagram of spaces over a fixed space $X$, where $I$ is a finite indexing category.
Let the objects in the diagram be called $Y_\alpha$ where $\alpha$
varies through the objects of $I$. Let
$$
Y = \text{hocolim}_{\alpha} Y_\alpha
$$
be the homotopy colimit of the diagram. Then $Y \in \text{Top}/X$ is an object.

Then Suppose $X' \to X$ is a map. Define $Z_\alpha$ to be the homotopy limit
$$
\text{holim}(X' \to X \leftarrow Y_{\alpha})
$$
This gives a diagram $I \to \text{Top}/X'$.

**Assertion:** The homotopy colimit of the diagram $\alpha \mapsto Z_\alpha$ coincides up to homotopy with the homotopy inverse limit of the diagram
$$
X' \to X \leftarrow Y \, .
$$
That is, homotopy colimits commute with base changes.

Your assertion follows from mine if we take $X'$ to be a point and $\alpha \mapsto Y_\alpha$ to be the diagram defining the wedge as a pushout.

As I recall, the assertion can be proved using a quasi-fibration argument. Another way is to use model categories.

Let's check a special case: suppose $I$ is the category $1 \leftarrow 12 \to 2$. The argument can be given in three steps:

Step 1: we can assume without loss in generality that $Y_1 \leftarrow Y_{12} \to Y_2$
is a diagram of Hurewicz fibrations over $X$ and that the arrows of the diagram are
cofibrations. In particular, the homotopy colimit of this diagram coincides with its colimit.

Step 2: According to a theorem of Arne Strøm, the base change of a cofibration is a cofibration. This means that the diagram over $X'$ of base changes
$$
Z_{1} \leftarrow Z_{12} \to Z_2
$$
consists of cofibrations, and each map to $X'$ is again a fibration.
Again, the homotopy colimit of this displayed diagram coincides with the colimit.

Step 3: The colimit of the diagram displayed in Step 2 is clearly the base change of
the colimit of the diagram displayed in Step 1.