= 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> $\xi\mathbin\#\eta$. 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 give
$$
\omega^\delta\leq\operatorname{otp}(X)\mathbin\#\operatorname{otp}(Y)<\omega^\delta,
$$
a contradiction. Thus at least one part has order type $\omega^\delta$.
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