Let $H$, $K$, $W$, be submodules of a module $M$.
Is it true that $(H \oplus K)/W \cong H/W \oplus K \cong H \oplus K/W$?
The first seems to follow from 1st isomorphism theorem on the map $\phi = \phi_{1} \oplus \phi_{2}: H \oplus K \rightarrow H/W \oplus K$, with $\phi_{1}$ the canonical map and $\phi_{2}$ the identity map. Similarly for the second isomorphism.
However, suppose $H, K \subset W$. Then it seems like $(H \oplus K)/W \cong 0$, contradicting the above. What am I missing?
Not necessarily. There are a number of fixes needed in your assumptions in order for the question to make more sense.
Especially of concern are issues of containment.
Where does the submodule $W$ exist in the module $M$ relative to $H$ and $K$? Is it contained in one of them?
For example, consider the $\mathbb{Z}$-module $\mathbb{Z}$ and submodules $H=2\mathbb{Z}$, $K=3\mathbb{Z}$ and $W=7\mathbb{Z}$.
$W$ is not contained in $H$ or $K$, so it doesn't make sense to take the quotient $(H\oplus K)/W$.
In other words, a quotient $M/T$ only makes sense if $T$ is a submodule of $M$.