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We study the stellar morphological evolution of disc galaxies within clusters in the TNG50 and TNG100 runs from the IllustrisTNG simulation suite. We select satellites of masses $10^{9.7} \leq M_{*,z=0}/\text{M}_{\odot} \leq 10^{11.6}$ residing in clusters of masses $10^{14} \lesssim M_{\text{200c,z=0}}/\text{M}_{\odot} \leq 10^{14.6}$ at $z=0$ and that were discs at accretion according to a kinematic morphology indicator (the circularity fraction). These are traced from the time of accretion to $z=0$ and compared to a control sample of central galaxies mass-matched at accretion. Most cluster discs become non-discy by $z=0$, in stark contrast with the control discs, of which a significant fraction remains discy over the same timescales. Cluster discs become non-discy accompanied by gas removal and star formation quenching, loss of dark matter and little growth or a loss of stellar mass. In contrast, control discs transform while also losing gas mass and quenching, but growing significantly in dark matter and stellar mass. Most cluster satellites change morphologies on similar timescales regardless of stellar mass, in $\sim0.5-4$ Gyr after accretion. Cluster discs that experienced more numerous and closer pericentric passages show the largest change in morphology. Morphological change in all cases requires the presence of a gravitational perturbation to drive stellar orbits to non-discy configurations, along with gas removal/heating to prevent replenishment of the disc through continued star-formation. For cluster discs, the perturbation is impulsive tidal shocking at pericentres and not tidal stripping of outer disc stellar material, whereas for control discs, a combination of mergers and AGN feedback appears to be the key driving force behind morphological transformations.