Basically, it is not a good idea to animate individual UV's because you have little control over the trajectory and speed. Technically you can, but the result.. will be inadequate to the time spent.
If you need just zooming out from the pattern as the mushroom grows, then animate the texture 2D placement node - there you have repetition and translation params - not so handy.
OR you can create a 3D projection node for that 2D texture, which will give even simpler control. Go Hypershade>Create>2DTextures>2DProjection switch, don't forget to switch it back to 2DNormal after you've done. this will create 3D placement node where you will be able to choose most appropriate projection type and transform+animate as needed.
The third option is to create an interesting animated texture in some 2D program, like AFX; just make it the growth animation length, or at least half of it; like, it could be some fractal mask that reveals the white cap cracks non-uniformely.
For the sake of control, I'd separate textures/shaders for the cap and the stem, or even make it two separate objects.
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Maya2019.1 @ Windows10 & GeForce GTX1080Ti
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