From Uniform to 2D gaussian
The Gaussian distribution is separable. Apply your transformation to each coordinate separately and you will get a 2D Gaussian.

If $G(x)$ is a 1D Gaussian, then $G(x) G(y)$ is a 2D Gaussian.

Thus, transform the distribution first according to one coordinate, creating a distribution that has normal distribution in one direction and uniform in the other. Then transform it again according to the other coordinate.