Can nuclear shielding affect relaxation time?
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.