jmc

When the point $(0,0)$ is reflected across the line $x=1$, it is reflected to point $(2,0)$ since the horizontal distance between the original point and the line is $1$. If we reflect the new point $(2,0)$ across the line $y=2$, the vertical distance between the point and the line is $2$, so the coordinates of the resulting point are $\boxed{(2,4)}$.