The Thom space is the homotopy cofiber of the Unit Sphere Projection:
$$ \mathbb{S}_\xi X \to X \to T(\xi) $$
while the unit sphere in a sumspace $\mathbb{S}[U\oplus V]$ is a join
$$ \begin{array}{c} \mathbb{S}[U] \times \mathbb{S}[V] & \rightarrow & \mathbb{S}[U] \\ \downarrow & & \downarrow \\
\mathbb{S}[V] & \rightarrow & \mathbb{S}[U\oplus V]
\end{array} $$ (homotopy-pushout). By one of "Mather's" Cube Theorems (it's a kind of distrutivity) this fiberwise pushout description integrates to a homotopy pushout of bundles:
$$ \begin{array}{c} \mathbb{S}_\xi \times_X \mathbb{S}_\eta &\rightarrow & \mathbb{S}_\xi \\ \downarrow & & \downarrow \\
\mathbb{S}_\eta & \rightarrow & \mathbb{S}_{\xi\oplus \eta}
\end{array} $$ The top and the left maps are both, as it happens, sphere fibrations, the ones relevant to your question. Consider the composable arrows
$$ \mathbb{S}_\xi \times_X \mathbb{S}_\eta \to \mathbb{S}_\xi \to X $$ and form the following small squares as homotopy pushouts:
$$ \begin{array}{c} \mathbb{S}_\xi \times_X \mathbb{S}_\eta &\rightarrow & \mathbb{S}_\xi & \rightarrow & \mathbb{S}_{\xi\oplus\eta} & \to & X \rlap{\simeq D\eta} \\ \downarrow & & \downarrow & & \downarrow & & \downarrow \\ * & \to & T(\eta|_{S\xi}) & \to
& T(\eta|_{S\xi}) \vee T(\xi|_{S\eta}) & \to & W \\ & & \downarrow & & \downarrow & & \downarrow \\
& & * & \to & T(\xi|_{S\eta}) & \to & T(\xi) \rlap{\simeq T(\xi|_{D\eta})}\\
&& && \downarrow & & \downarrow \\ & & & & * & \to & T(\xi\oplus\eta) \end{array} $$ where $W$ doesn't matter for the rest of the argument.

Now, I've sneakily applied the homotopy pushout pasting lemma, several times: to identify the $-\vee-$ in the second row (which gives the Thom space below it) and to identify the other Thom spaces we want. That says the homotopy type we want is the homotopy cofiber of a map of Thom spaces; and the particular fine result you want is: the inclusion $T(\xi|_{S\eta}) \to T(\xi|_{D\eta})$ is a **cofibration** and represents the homotopy class of the pushout structure map indicated by the diagram.