It seems there's no significance in the fact that the galactic plane is tilted relative to the plane of the solar system and relative to the plane of Earth's equator, and no reason to predict that the plane of the solar system should match the plane of the galaxy. If we look around at multiple star systems, we see the planes of their orbits are arranged at all sorts of angles to the galactic plane; it seems that what must influence the plane of rotation of a star system is the random motion within the cloud it forms from, rather than the rotation of the galaxy as a whole.
So: the tilt of the Milky Way in the sky isn't telling us anything significant about the solar system's origin.

As to the axis of Uranus, it turns out not to be aligned with the galaxy's axis: the galactic north pole points towards the constellation Coma Bernices; Uranus' north pole points between Orion and Taurus. The separation between the two axes must be around 120 degrees.

Grant Hutchison