= Solution
The two standard <affine scheme>[affine] charts of the blowup restrict to an affine open cover of $X_0$. On the $x$-chart write $y=xu$. Since $xy=x^2u$, the fiber is
$$
U_x\cap X_0=\operatorname{Spec}k[x,u]/(x^2u).
$$
On the $y$-chart write $x=yv$, giving
$$
U_y\cap X_0=\operatorname{Spec}k[y,v]/(y^2v).
$$
On their overlap, $u$ and $v$ are invertible with $v=u^{-1}$ and $y=xu$. The squared factors $x^2$ and $y^2$ display the doubled exceptional component explicitly.
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