Can nuclear shielding affect relaxation time? Can nuclear shielding affect the transverse relaxation time, i.e extend the relaxation time the more the nuclei is shielded?
I'd say yes, but not by any detectable amount. In some relaxation mechanisms (e.g. dipolar relaxation, a major relaxation pathway for $\ce{^1H}$), the rate of relaxation varies with $\langle B_\mathrm{loc}^2\rangle$, i.e. the magnitude of the random fields that the spin experiences due to the magnetic momenta of all the other spins. The magnitude of any field that a nucleus experiences will be attenuated by the induced magnetic field arising from its own electrons, so it stands to reason that – all else being equal – a more shielded nucleus will have a smaller $\langle B_\mathrm{loc}^2\rangle$, and a longer relaxation time, than a less shielded nucleus. This applies to both $T_1$ and $T_2$.

Bear in mind, however, the most obvious effect of nuclear shielding, which is to change the resonance frequency of the nucleus. The spread of frequencies is on the order of $10^{-6}$ relative to the strength of the external magnetic field. If we crudely estimate that the spread in $T_1$ and $T_2$ arising from sheilding differences is also on the ppm level, that corresponds to ~µs levels of difference, which I strongly doubt is detectable.

In any case, for two protons that have different chemical environments, there would be other factors which affect the relaxation times to a much greater extent.