 Solution

Begin with the product cobordism $(X_0\bigsqcup X_1)\times [0,1]$. 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,This is the one-handle cobordism from a disjoint union to a connected sum.