This seems to be answered negatively in "On the composition and products of universal mappings" by W. Holsztyński (Fundamenta Mathematicae 64(2) (1969), 181-188).

I've not looked at the proof or really tried to figure out why it works, but if I understand correctly what is claimed, then one counterexample from the paper is as follows:

$M$ is a Möbius band, regarded as the annulus
$\left\{z\in\mathbb{C}:\frac{1}{2}\leq\vert z\vert\leq 1\right\}$
with $z$ and $-z$ identified for $\vert z\vert=\frac{1}{2}$.

$Q$ is the closed unit disc $\left\{z\in\mathbb{C}:\vert z\vert\leq 1\right\}$.

Apparently the map $u:M\to Q$ induced by $z\mapsto\left(2-\frac{1}{\vert z\vert}\right)z$, which identifies the inner boundary of the annulus to a point, and the map $v:Q\to Q$ given by $z\mapsto z^2$ are both universal, but $v\circ u$ is not.