Does Mersenne Twister generate homogeneous numbers?
I think the Mersenne twister satisfies the requirement for homogeneity, since it produces numbers that are equally distributed (with 32-bit accuracy) in up to 623 dimensions.

But in a cryptography context, it could hardly be described as "uniform", because you only need to observe 624 outputs in order to predict all the following values with 100% accuracy.

I would recommend using a cryptographically secure PRNG such as AES-CTR instead. This should provide you with at least 100 MB of random data per second.