Does the axis of a galaxy as a whole "wobble" the same way that the Earth does? If so do we have any idea how much or how fast the Milky Way is doing so?
To expand a bit on FJC's answer: galactic disks are not rigid bodies, and so we should not expect them to rigidly precess.
Most disk galaxies do show warps at large radii (including our own), in both the interstellar gas and the stars. From observations of warps in gas disks, we know that the azimuthal angle of the warp often changes with radius: i.e., the tilt of the disk changes as a function of radius.[*] Since the angular rotation speed changes as a function of radius,[**] this suggests the tilt could precess at different rates for different radii.
Unfortunately, since there isn't really a consensus on what causes or maintains warps, it's very hard to go from observing a warp to predicting what the underlying precession rates might or might not be. (Although one can argue that the precession rate can't vary too much with radius, otherwise the warp would rapidly "wind up" and lose its coherent warped shape, and we wouldn't see warps as often as we do.)
See here for some discussion of the theoretical issues.
[*] This is a kind of precession, although it's a change in the orientation of the rotational axis with radius rather than a change with time (as in the Earth's precession).
[**] Even if the circular velocity is approximately constant, as it often is in the outer regions of galaxies, the increase in the circumference as a function of increasing radius means the angular velocity goes down.
The whole galactic disk is made up of many individual stars and the interstellar gas between them. The whole thing doesn't tilt or wobble as a solid object would.
Other parts of the Galaxy do precess though. The central supermassive black hole might and therefore the jest coming out of it might move around like the axis of the Earth's rotation does.
Also, galactic disks do wobble but not like a solid disk, more like a series of rings - the orbits are different at different distances from the centre. These can change very slowly and parts of the disk can be as a different angle than other parts.
This is generally not well understood because we can't see the disks very well, being so far away, and they would change very slowly.
Precession is a general property of rotating systems under forces from external sources. For instance, Earth's precession is due to gravitational force from Sun and Moon. Since the galaxy is very far away from any other gravitational source, I don't see a reason for its precession.
Edit: as pointed out in the comment below (@RobJeffries), if it is not sufficiently far from massive system, than it will possibly precess...