I tried deriving distributions of numerator and denominator separately. But found that there is no closed form. I have no clue on how to show that Z is standard normal.
Consider the conditional distribution $Z\mid( U_1=u_1,U_2=u_2)$ and use what you know about the reproductive property of univariate normal distribution. You would find that $Z\mid U_1,U_2$ is independent of $U_1,U_2$.