Do gas giants wobble, or it that just reserved to terrestrial planets?
Yes - The spin axes of Jupiter precess with a precessional time period of
All spinning objects that are not perfectly rigid will develop equatorial bulge which spoils the spherical symmetry. Because of this equatorial bulge the planet's gravitational interaction with Sun and its satellites (moons) could result in a torque that would lead to precession in its spin axis.
The net precessional torque will be the sum of the torque due to the Sun's interaction and the torques due to the interaction of all its satellites (moons).
If the planet's spin axis is exactly perpendicular to its ecliptic plane (orbital plane) then Sun's interaction would not exert any torque. However if the planet's spin axis is tilted it would result in a torque.
Similarly if the satellite is orbiting the planet in its equatorial plane then its interaction with equatorial bulge will not generate a torque. If the satellite is in a non equatorial orbit it would generate a torque.