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Everywhere in the literature I have across, I find that whenever the discussion of the Induced electric field comes, it is discussed around a cylindrical time-varying magnetic field. The induced electric field turns out to be circular in this case.

My question is, what shape does the induced electric field take when the magnetic field is time-varying but non-cylindrical?

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Electric and magnetic field around a coil of rectangular cross section. Figure 1 shows the geometry of the coil, the distribution of current (red) and magnetic field (blue). Figure 1

In fig. 2 shows the distribution of the magnetic field in various sections $y=0,0.2, 0.4, 0.6, 0.8, 1$ Figure 2

In fig. 3 shows the distribution of the electric field in various sections $z=0,0.2, 0.4, 0.6, 0.8, 1$.It is seen that when moving away from the center of the coil, the lines of the electric field become circular as @garyp noted.

Figure3

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The field lines would close as they do for a circle, but the shape would not be circular. The exact shape depends on the distribution of the electric field.

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    $\begingroup$ Thanks a lot for the response. Could you kindly put some links to some resources or some images which expound on the idea further? Some example images on how the electric field lines will look like in some sample scenarios? $\endgroup$ – Devansh Mittal Nov 22 '19 at 18:11

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