Which vectors have unique representation when it isn't a direct sum?

If $V=U+W$ is not direct, then there is $z \in U \cap W$ with $z \ne 0.$

Let $v \in V,$ then there are $u \in U$ and $w \in W$ such that

$$v=u+w.$$

We also have

$$v=(u-z)+(w+z).$$

Observe that $u-z \in U, w+z \in W, u-z \ne u$ and $w+z \ne w.$

Consequence: each $v \in V$ does not have a unique representation.

If one vector can be written in more than one way, then all can. Let $v=u+w$ for different pairs $\{u,w\}$ and write $x=(x-v)+v=(u'+w')+(u+w)=(u'+u)+(w'+w)$.