Uniformly at random vs. randomly uniformly

The phrase "uniformly at random" is a very common phrase in probability theory, and people in the field will understand it, even if it isn't precise if you read it as an ordinary English sentence.

If you want to replace it, you should use "randomly and uniformly", which is also a very common phrase in probability theory. Why do you need an "and"? These two adjectives have the same place in natural adjective order, so they are coordinate adjectives and not cumulative adjectives; this means they want an "and" to separate them. (And furthermore, the mathematical community usually uses an "and" here.) See this question about using "and" between adjectives.

If you're worried that your readers might not understand "uniformly at random", there is nothing wrong with replacing it with "randomly and uniformly".

Here is a Google Ngram in support of these statements.