By weak mixing you mean that product dynamical system is mixing, right? Then the answer is yes. See for instance Proposition 2.4, point (3) of this 2015 paper.

For the proof, the authors refer to Theorem 1.11 in: E. Glasner, Ergodic theory via joinings, Mathematical Surveys and Monographs, 101. American Mathematical Society, 2003.

By weak mixing you mean that product dynamical system is topologically mixing, right? Then the answer is yes, and in fact the two facts are equivalent. See for instance Proposition 2.4, point (3) of this 2015 paper.

For the proof, the authors refer to Theorem 1.11 in: E. Glasner, Ergodic theory via joinings, Mathematical Surveys and Monographs, 101. American Mathematical Society, 2003.

(Of course I'm assuming that YCor is right in his comment and that $U,V$ are nonempty open sets.)