Effect on T1 relaxation due to the application of the gradient
It does not have any practical effect on the $T_1$ relaxation, if there is a gradient present or not. 

Sure, from an academic point of view, there is an effect: Typically, in biomedical applications $T_1$ gets longer at higher field strengths. This is the reason why imaging sequences have to be adapted to the field strength of the machine, if $T_1$-dependent signal behavior is exploited (see Fluid Attenuation, or magnetization prearation in MPRAGE as an example). So, since you increase the magnetic field in $z$ diretion by a few mT, this would also influence $T_1$. However, I have never seen a practical application where this was a concern.