 Solution

Use the shuffle bound: if a well-order is partitioned into subsets of order types $\xi$ and $\eta$, its order type is at most the Hessenberg natural sum $\xi \#\eta$. This follows by merging the two induced well-orders and comparing their Cantor-normal-form terms.

If both $X$ and $Y$ had order type strictly below ${\omega }^{\delta }$, each would have leading Cantor exponent below $\delta$. Their natural sum would still be below ${\omega }^{\delta }$. The shuffle bound would then givea contradiction. Thus at least one part has order type ${\omega }^{\delta }$.