First, a point of terminology: uncountably infinite refers to any infinite cardinality other than countably infinite, so if the number of probability distributions is at least the cardinality of the real numbers, then it's already uncountable.
As for what uncountable cardinality they form, it's the same as the real numbers [0]. Roughly, a probability distribution is determined by the countable collection of real numbers P(X <= q) with q rational. That means the cardinality is no more than R^Q, which is isomorphic to R. (the cardinality is at least R due to, say, uniform distributions on [0, x] for x real).
[0] https://math.stackexchange.com/questions/3698864/why-does-th...