= Solution
Begin with the product cobordism $(X_0\sqcup X_1)\times[0,1]$. In its outgoing boundary choose one embedded $n$-ball in each component and attach an $(n+1)$-dimensional one-handle $D^1\times D^n$ along $S^0\times D^n$. On the outgoing boundary this deletes those two balls and joins their boundary spheres by $D^1\times S^{n-1}$, which is precisely the connected-sum construction. With the product orientation,
$$
\boxed{\partial W=-(X_0\sqcup X_1)\sqcup(X_0\mathbin\#X_1).}
$$
This is the <one-handle cobordism from a disjoint union to a connected sum>.
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