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)} $.