Recall that $\mathbb{RP}^2\#\mathbb{RP}^2$ is the Klein bottle and can be seen as a non-trivial $S^1$-bundle over $S^1$. In particular, it is the total space of the sphere bundle of $\gamma\oplus\varepsilon^1$ over $S^1$ where $\gamma$ is the Möbius line bundle and $\varepsilon^1$ is the trivial line bundle; note, $\gamma\oplus \varepsilon^1$ is the unique non-trivial rank two vector bundle on $S^1$ (because $\pi_0(O(2)) = \mathbb{Z}_2$) and it is not orientable (i.e. $w_1(\gamma\oplus\varepsilon^1) \neq 0$) as is $K$.

Also $\mathbb{CP}^2\#\overline{\mathbb{CP}^2}$ can be seen as a non-trivial $S^2$-bundle over $S^2$. One way to do this is to view it as the first Hirzebruch surface $\mathbb{P}(\mathcal{O}(-1)\oplus\mathcal{O})$; here $\mathcal{O}(-1)$ and $\mathcal{O}$ are holomorphic line bundles over $\mathbb{CP}^1$. Alternatively, it is the total space of the sphere bundle of $\mathcal{O}(-1)\oplus\varepsilon^1$ over $S^2$. Note that $\mathcal{O}(-1)\oplus\varepsilon^1$ is the unique non-trivial rank three vector bundle on $S^2$ (because $\pi_1(SO(3)) = \mathbb{Z}_2$) and it is not spin (i.e. $w_2(\mathcal{O}(1)\oplus\varepsilon^1)) \neq 0$) as is $\mathbb{CP}^2\#\overline{\mathbb{CP}^2}$.

This naturally leads one to ask:

Can $\mathbb{HP}^2\#\overline{\mathbb{HP}^2}$ and $\mathbb{OP}^2\#\overline{\mathbb{OP}^2}$ be realised as (non-trivial) sphere bundles over $S^4$ and $S^8$ respectively?

Not all sphere bundles in these dimensions necessarily arise as sphere bundles of vector bundles. However, given the constructions above, I would expect them to in this case.

Unlike in the real and complex cases, there is not a unique non-trivial vector bundle to take the sphere bundle of because $\pi_3(SO(5)) = \mathbb{Z}$ and $\pi_7(SO(9)) = \mathbb{Z}$. All of these homotopy groups are in the stable range, so it is no coincidence that there we have two $\mathbb{Z}_2$'s and two $\mathbb{Z}$'s appearing. As $w_4(\mathbb{HP}^2\#\overline{\mathbb{HP}^2}) \neq 0$ and $w_8(\mathbb{OP}^2\#\overline{\mathbb{OP}^2}) \neq 0$, the desired vector bundles should have non-zero $w_4$ and $w_8$ respectively (in direct analogy with the cases above).

Note that both $\gamma$ and $\mathcal{O}(-1)$ can be viewed as the tautological (real/complex) line bundles over $\mathbb{RP}^1$ and $\mathbb{CP}^1$ respectively. This suggests that we should try the direct sum of the tautological (quaternionic/octionic) line bundles over $\mathbb{HP}^2$ and $\mathbb{OP}^2$ with $\varepsilon^1$. This is perfectly reasonable in the quaternionic case, but $\mathbb{OP}^2$ does not arise as the projectivisation of $\mathbb{O}^3$, so there is no such bundle; maybe there is a natural replacelement though (probably associated to one of the generators of $\pi_7(SO(8)) \cong \mathbb{Z}\oplus\mathbb{Z}$).